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OCR C2 2005 January Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).
  1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
  2. Find the exact area of the shaded region.
OCR C2 2005 January Q8
9 marks Standard +0.3
8
  1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
    1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
    2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
    3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$
OCR C2 2005 January Q9
11 marks Standard +0.3
9 A geometric progression has first term \(a\), where \(a \neq 0\), and common ratio \(r\), where \(r \neq 1\). The difference between the fourth term and the first term is equal to four times the difference between the third term and the second term.
  1. Show that \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1 = 0\).
  2. Show that \(r - 1\) is a factor of \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\). Hence factorise \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\).
  3. Hence find the two possible values for the ratio of the geometric progression. Give your answers in an exact form.
  4. For the value of \(r\) for which the progression is convergent, prove that the sum to infinity is \(\frac { 1 } { 2 } a ( 1 + \sqrt { } 5 )\).
OCR MEI S4 2007 June Q1
24 marks Challenging +1.2
1 The random variable \(X\) has the continuous uniform distribution with probability density function $$\mathrm { f } ( x ) = \frac { 1 } { \theta } , \quad 0 \leqslant x \leqslant \theta$$ where \(\theta ( \theta > 0 )\) is an unknown parameter.
A random sample of \(n\) observations from \(X\) is denoted by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\), with sample mean \(\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).
  1. Show that \(2 \bar { X }\) is an unbiased estimator of \(\theta\).
  2. Evaluate \(2 \bar { X }\) for a case where, with \(n = 5\), the observed values of the random sample are \(0.4,0.2\), 1.0, 0.1, 0.6. Hence comment on a disadvantage of \(2 \bar { X }\) as an estimator of \(\theta\). For a general random sample of size \(n\), let \(Y\) represent the sample maximum, \(Y = \max \left( X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right)\). You are given that the probability density function of \(Y\) is $$g ( y ) = \frac { n y ^ { n - 1 } } { \theta ^ { n } } , \quad 0 \leqslant y \leqslant \theta$$
  3. An estimator \(k Y\) is to be used to estimate \(\theta\), where \(k\) is a constant to be chosen. Show that the mean square error of \(k Y\) is $$k ^ { 2 } \mathrm { E } \left( Y ^ { 2 } \right) - 2 k \theta \mathrm { E } ( Y ) + \theta ^ { 2 }$$ and hence find the value of \(k\) for which the mean square error is minimised.
  4. Comment on whether \(k Y\) with the value of \(k\) found in part (iii) suffers from the disadvantage identified in part (ii).
OCR MEI S4 2007 June Q2
24 marks Challenging +1.2
2 The random variable \(X\) has the binomial distribution with parameters \(n\) and \(p\), i.e. \(X \sim \mathrm {~B} ( n , p )\).
  1. Show that the probability generating function of \(X\) is \(\mathrm { G } ( t ) = ( q + p t ) ^ { n }\), where \(q = 1 - p\).
  2. Hence obtain the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of \(X\).
  3. Write down the mean and variance of the random variable \(Z = \frac { X - \mu } { \sigma }\).
  4. Write down the moment generating function of \(X\) and use the linear transformation result to show that the moment generating function of \(Z\) is $$\mathrm { M } _ { Z } ( \theta ) = \left( q \mathrm { e } ^ { - \frac { p \theta } { \sqrt { n p q } } } + p \mathrm { e } ^ { \frac { q \theta } { \sqrt { n p q } } } \right) ^ { n } .$$
  5. By expanding the exponential terms in \(\mathrm { M } _ { Z } ( \theta )\), show that the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(n \rightarrow \infty\) is \(\mathrm { e } ^ { \theta ^ { 2 } / 2 }\). You may use the result \(\lim _ { n \rightarrow \infty } \left( 1 + \frac { y + \mathrm { f } ( n ) } { n } \right) ^ { n } = \mathrm { e } ^ { y }\) provided \(\mathrm { f } ( n ) \rightarrow 0\) as \(n \rightarrow \infty\).
  6. What does the result in part (v) imply about the distribution of \(Z\) as \(n \rightarrow \infty\) ? Explain your reasoning briefly.
  7. What does the result in part (vi) imply about the distribution of \(X\) as \(n \rightarrow \infty\) ?
OCR MEI S4 2007 June Q3
24 marks Challenging +1.2
3 An engineering company buys a certain type of component from two suppliers, A and B. It is important that, on the whole, the strengths of these components are the same from both suppliers. The company can measure the strengths in its laboratory. Random samples of seven components from supplier A and five from supplier B give the following strengths, in a convenient unit.
Supplier A25.827.426.223.528.326.427.2
Supplier B25.624.923.725.826.9
The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.
  1. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
  2. Provide a two-sided 90\% confidence interval for the true mean difference.
  3. Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
  4. A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.
OCR MEI S4 2007 June Q4
24 marks Standard +0.8
4 An agricultural company conducts a trial of five fertilisers (A, B, C, D, E) in an experimental field at its research station. The fertilisers are applied to plots of the field according to a completely randomised design. The yields of the crop from the plots, measured in a standard unit, are analysed by the one-way analysis of variance, from which it appears that there are no real differences among the effects of the fertilisers. A statistician notes that the residual mean square in the analysis of variance is considerably larger than had been anticipated from knowledge of the general behaviour of the crop, and therefore suspects that there is some inadequacy in the design of the trial.
  1. Explain briefly why the statistician should be suspicious of the design.
  2. Explain briefly why an inflated residual leads to difficulty in interpreting the results of the analysis of variance, in particular that the null hypothesis is more likely to be accepted erroneously. Further investigation indicates that the soil at the west side of the experimental field is naturally more fertile than that at the east side, with a consistent 'fertility gradient' from west to east.
  3. What experimental design can accommodate this feature? Provide a simple diagram of the experimental field indicating a suitable layout. The company decides to conduct a new trial in its glasshouse, where experimental conditions can be controlled so that a completely randomised design is appropriate. The yields are as follows.
    Fertiliser AFertiliser BFertiliser CFertiliser DFertiliser E
    23.626.018.829.017.7
    18.235.316.737.216.5
    32.430.523.032.612.8
    20.831.428.331.420.4
    [The sum of these data items is 502.6 and the sum of their squares is 13610.22 .]
  4. Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a \(5 \%\) significance level. Report briefly on your conclusions.
  5. State the assumptions about the distribution of the experimental error that underlie your analysis in part (iv).
OCR C2 2006 January Q1
6 marks Moderate -0.8
1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .
  1. Find the first term and the common difference.
  2. Show that the sum of the first 29 terms is zero.
OCR C2 2006 January Q2
6 marks Moderate -0.8
2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate
  1. the area of the triangle,
  2. the length of \(C A\),
  3. the size of angle \(C\).
OCR C2 2006 January Q3
6 marks Moderate -0.8
3
  1. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$
OCR C2 2006 January Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The angle \(A O B\) is 1.8 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. It is given that \(O A = O B = 20 \mathrm {~cm}\) and \(O C = O D = 15 \mathrm {~cm}\). The shaded region is bounded by the arcs \(A B\) and \(C D\) and by the lines \(C A\) and \(D B\).
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
OCR C2 2006 January Q5
8 marks Standard +0.3
5 In a geometric progression, the first term is 5 and the second term is 4.8 .
  1. Show that the sum to infinity is 125 .
  2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).
OCR C2 2006 January Q6
8 marks Standard +0.3
6
  1. Find \(\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x\).
    1. Find the value, in terms of \(a\), of \(\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x\), where \(a\) is a constant greater than 1 .
    2. Deduce the value of \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\).
OCR C2 2006 January Q7
8 marks Moderate -0.5
7
  1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
    1. \(\log _ { 10 } \left( \frac { x } { y } \right)\)
    2. \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
    3. Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of \(y\) correct to 3 decimal places.
OCR C2 2006 January Q8
12 marks Moderate -0.3
8 The cubic polynomial \(2 x ^ { 3 } + k x ^ { 2 } - x + 6\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  1. Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
  2. Find \(\int _ { - 1 } ^ { 2 } f ( x ) \mathrm { d } x\).
  3. Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 1 \leqslant x \leqslant 2\). \section*{[Question 9 is printed overleaf.]}
OCR MEI S4 2008 June Q1
24 marks Challenging +1.2
1 The random variable \(X\) has the Poisson distribution with parameter \(\theta\) so that its probability function is $$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \theta } \theta ^ { x } } { x ! } , \quad x = 0,1,2 , \ldots$$ where \(\theta ( \theta > 0 )\) is unknown. A random sample of \(n\) observations from \(X\) is denoted by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\).
  1. Find \(\hat { \theta }\), the maximum likelihood estimator of \(\theta\). The value of \(\mathrm { P } ( X = 0 )\) is denoted by \(\lambda\).
  2. Write down an expression for \(\lambda\) in terms of \(\theta\).
  3. Let \(R\) denote the number of observations in the sample with value zero. By considering the binomial distribution with parameters \(n\) and \(\mathrm { e } ^ { - \theta }\), write down \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\). Deduce that the observed proportion of observations in the sample with value zero, denoted by \(\tilde { \lambda }\), is an unbiased estimator of \(\lambda\) with variance \(\frac { \mathrm { e } ^ { - \theta } \left( 1 - \mathrm { e } ^ { - \theta } \right) } { n }\).
  4. In large samples, the variance of the maximum likelihood estimator of \(\lambda\) may be taken as \(\frac { \theta \mathrm { e } ^ { - 2 \theta } } { n }\). Use this and the appropriate result from part (iii) to show that the relative efficiency of \(\tilde { \lambda }\) with respect to the maximum likelihood estimator is \(\frac { \theta } { \mathrm { e } ^ { \theta } - 1 }\). Show that this expression is always less than 1 . Show also that it is near 1 if \(\theta\) is small and near 0 if \(\theta\) is large.
OCR MEI S4 2008 June Q2
24 marks Standard +0.8
2 Independent trials, on each of which the probability of a 'success' is \(p ( 0 < p < 1 )\), are being carried out. The random variable \(X\) counts the number of trials up to and including that on which the first success is obtained. The random variable \(Y\) counts the number of trials up to and including that on which the \(n\)th success is obtained.
  1. Write down an expression for \(\mathrm { P } ( X = x )\) for \(x = 1,2 , \ldots\). Show that the probability generating function of \(X\) is $$\mathrm { G } ( t ) = p t ( 1 - q t ) ^ { - 1 }$$ where \(q = 1 - p\), and hence that the mean and variance of \(X\) are $$\mu = \frac { 1 } { p } \quad \text { and } \quad \sigma ^ { 2 } = \frac { q } { p ^ { 2 } }$$ respectively.
  2. Explain why the random variable \(Y\) can be written as $$Y = X _ { 1 } + X _ { 2 } + \ldots + X _ { n }$$ where the \(X _ { i }\) are independent random variables each distributed as \(X\). Hence write down the probability generating function, the mean and the variance of \(Y\).
  3. State an approximation to the distribution of \(Y\) for large \(n\).
  4. The aeroplane used on a certain flight seats 140 passengers. The airline seeks to fill the plane, but its experience is that not all the passengers who buy tickets will turn up for the flight. It uses the random variable \(Y\) to model the situation, with \(p = 0.8\) as the probability that a passenger turns up. Find the probability that it needs to sell at least 160 tickets to get 140 passengers who turn up. Suggest a reason why the model might not be appropriate.
OCR MEI S4 2008 June Q3
24 marks Standard +0.3
3
  1. Explain the meaning of the following terms in the context of hypothesis testing: Type I error, Type II error, operating characteristic. A machine fills salt containers that will be sold in shops. The containers are supposed to contain 750 g of salt. The machine operates in such a way that the amount of salt delivered to each container is a Normally distributed random variable with standard deviation 20 g . The machine should be calibrated in such a way that the mean amount delivered, \(\mu\), is 750 g . Each hour, a random sample of 9 containers is taken from the previous hour's output and the sample mean amount of salt is determined. If this is between 735 g and 765 g , the previous hour's output is accepted. If not, the previous hour's output is rejected and the machine is recalibrated.
  2. Find the probability of rejecting the previous hour's output if the machine is properly calibrated. Comment on your result.
  3. Find the probability of accepting the previous hour's output if \(\mu = 725 \mathrm {~g}\). Comment on your result.
  4. Obtain an expression for the operating characteristic of this testing procedure in terms of the cumulative distribution function \(\Phi ( z )\) of the standard Normal distribution. Evaluate the operating characteristic for the following values (in g) of \(\mu\) : 720, 730, 740, 750, 760, 770, 780.
OCR MEI S4 2008 June Q4
24 marks Standard +0.3
4
  1. State the usual model, including the accompanying distributional assumptions, for the one-way analysis of variance. Interpret the terms in the model.
  2. An examinations authority is considering using an external contractor for the typesetting and printing of its examination papers. Four contractors are being investigated. A random sample of 20 examination papers over the entire range covered by the authority is selected and 5 are allocated at random to each contractor for preparation. The authority carefully checks the printed papers for errors and assigns a score to each to indicate the overall quality (higher scores represent better quality). The scores are as follows.
    Contractor AContractor BContractor CContractor D
    41545641
    49454536
    50505446
    44505038
    56474935
    [The sum of these data items is 936 and the sum of their squares is 44544 .]
    Construct the usual one-way analysis of variance table. Carry out the appropriate test, using a \(5 \%\) significance level. Report briefly on your conclusions.
  3. The authority thinks that there might be differences in the ways the contractors cope with the preparation of examination papers in different subject areas. For this purpose, the subject areas are broadly divided into mathematics, sciences, languages, humanities, and others. The authority wishes to design a further investigation, ensuring that each of these subject areas is covered by each contractor. Name the experimental design that should be used and describe briefly the layout of the investigation.
OCR MEI S4 2010 June Q1
24 marks Standard +0.8
1 The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \frac { x \mathrm { e } ^ { - x / \lambda } } { \lambda ^ { 2 } } \quad ( x > 0 )$$ where \(\lambda\) is a parameter \(( \lambda > 0 ) . X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) are \(n\) independent observations on \(X\), and \(\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\) is their mean.
  1. Obtain \(\mathrm { E } ( X )\) and deduce that \(\hat { \lambda } = \frac { 1 } { 2 } \bar { X }\) is an unbiased estimator of \(\lambda\).
  2. \(\operatorname { Obtain } \operatorname { Var } ( \hat { \lambda } )\).
  3. Explain why the results in parts (i) and (ii) indicate that \(\hat { \lambda }\) is a good estimator of \(\lambda\) in large samples.
  4. Suppose that \(n = 3\) and consider the alternative estimator $$\tilde { \lambda } = \frac { 1 } { 8 } X _ { 1 } + \frac { 1 } { 4 } X _ { 2 } + \frac { 1 } { 8 } X _ { 3 } .$$ Show that \(\tilde { \lambda }\) is an unbiased estimator of \(\lambda\). Find the relative efficiency of \(\tilde { \lambda }\) compared with \(\hat { \lambda }\). Which estimator do you prefer in this case?
OCR MEI S4 2010 June Q2
24 marks Standard +0.8
2 The random variable \(X\) has the Poisson distribution with parameter \(\lambda\).
  1. Show that the probability generating function of \(X\) is \(\mathrm { G } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }\).
  2. Hence obtain the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of \(X\).
  3. Write down the mean and variance of the random variable \(Z = \frac { X - \mu } { \sigma }\).
  4. Write down the moment generating function of \(X\). State the linear transformation result for moment generating functions and use it to show that the moment generating function of \(Z\) is $$\mathrm { M } _ { Z } ( \theta ) = \mathrm { e } ^ { \mathrm { f } ( \theta ) } \quad \text { where } \mathrm { f } ( \theta ) = \lambda \left( \mathrm { e } ^ { \theta / \sqrt { \lambda } } - \frac { \theta } { \sqrt { \lambda } } - 1 \right)$$
  5. Show that the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(\lambda \rightarrow \infty\) is \(\mathrm { e } ^ { \theta ^ { 2 } / 2 }\).
  6. Explain briefly why this implies that the distribution of \(Z\) tends to \(\mathrm { N } ( 0,1 )\) as \(\lambda \rightarrow \infty\). What does this imply about the distribution of \(X\) as \(\lambda \rightarrow \infty\) ?
OCR MEI S4 2010 June Q3
24 marks Standard +0.3
3 At a factory, two production lines are in use for making steel rods. A critical dimension is the diameter of a rod. For the first production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.2 mm . For the second production line, it is assumed from experience that the diameters are Normally distributed with standard deviation 1.4 mm . It is desired to test whether the mean diameters for the two production lines, \(\mu _ { 1 }\) and \(\mu _ { 2 }\), are equal. A random sample of 8 rods is taken from the first production line and, independently, a random sample of 10 rods is taken from the second production line.
  1. Find the acceptance region for the customary test based on the Normal distribution for the null hypothesis \(\mu _ { 1 } = \mu _ { 2 }\), against the alternative hypothesis \(\mu _ { 1 } \neq \mu _ { 2 }\), at the \(5 \%\) level of significance.
  2. The sample means are found to be 25.8 mm and 24.4 mm respectively. What is the result of the test? Provide a two-sided \(99 \%\) confidence interval for \(\mu _ { 1 } - \mu _ { 2 }\). The production lines are modified so that the diameters may be assumed to be of equal (but unknown) variance. However, they may no longer be Normally distributed. A two-sided test of the equality of the population medians is required, at the \(5 \%\) significance level.
  3. The diameters in independent random samples of sizes 6 and 8 are as follows, in mm .
    First production line25.925.825.324.724.425.4
    Second production line23.825.624.023.524.124.524.325.1
    Use an appropriate procedure to carry out the test.
OCR MEI S4 2010 June Q4
24 marks Standard +0.3
4 At an agricultural research station, a trial is made of four varieties (A, B, C, D) of a certain crop in an experimental field. The varieties are grown on plots in the field and their yields are measured in a standard unit.
  1. It is at first thought that there may be a consistent trend in the natural fertility of the soil in the field from the west side to the east, though no other trends are known. Name an experimental design that should be used in these circumstances and give an example of an experimental layout. Initial analysis suggests that any natural fertility trend may in fact be ignored, so the data from the trial are analysed by one-way analysis of variance.
  2. The usual model for one-way analysis of variance of the yields \(y _ { i j }\) may be written as $$y _ { i j } = \mu + \alpha _ { i } + e _ { i j }$$ where the \(e _ { i j }\) represent the experimental errors. Interpret the other terms in the model. State the usual distributional assumptions for the \(e _ { i j }\).
  3. The data for the yields are as follows, each variety having been used on 5 plots.
    Variety
    ABCD
    12.314.214.113.6
    11.913.113.212.8
    12.813.114.613.3
    12.212.513.714.3
    13.512.713.413.8
    $$\left[ \Sigma \Sigma y _ { i j } = 265.1 , \quad \Sigma \Sigma y _ { i j } ^ { 2 } = 3524.31 . \right]$$ Construct the usual one-way analysis of variance table and carry out the usual test, at the 5\% significance level. Report briefly on your conclusions. {www.ocr.org.uk} after the live examination series.
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OCR C2 2008 January Q1
4 marks Moderate -0.3
1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram.
OCR C2 2008 January Q2
4 marks Easy -1.2
2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of $$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$