1.08d Evaluate definite integrals: between limits

7 \includegraphics[max width=\textwidth, alt={}, center]{78a9b100-c3bd-4054-b539-ec8304440063-10_551_641_260_751} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3 ,$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and the shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\).

1. Find the exact \(x\)-coordinate of \(A\).
2. Find the exact gradient of the curve at \(A\).
3. Find the exact area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Find the exact value of \(\int _ { - 1 } ^ { 2 } \left( 4 \mathrm { e } ^ { 2 x } - 2 \mathrm { e } ^ { - x } \right) \mathrm { d } x\).

4 Given that \(\int _ { a } ^ { a + 14 } \frac { 1 } { 3 x } \mathrm {~d} x = \ln 2\), find the value of the positive constant \(a\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 12 x ^ { 3 } - 9 x ^ { 2 } + 8 x - 4$$

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by \(( 4 x - 3 )\) and show that the remainder is 2 .
2. Hence find \(\int _ { 2 } ^ { 12 } \left( \frac { \mathrm { p } ( x ) } { 4 x - 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\), giving your answer in the form \(a + \ln b\).

3 \includegraphics[max width=\textwidth, alt={}, center]{a31a4b4e-83a6-47d9-9679-3471b3da1b6e-2_488_664_863_737} The diagram shows the curve \(y = \mathrm { e } ^ { 2 x }\). The shaded region \(R\) is bounded by the curve and by the lines \(x = 0 , y = 0\) and \(x = p\).

1. Find, in terms of \(p\), the area of \(R\).
2. Hence calculate the value of \(p\) for which the area of \(R\) is equal to 5 . Give your answer correct to 2 significant figures.

6 The equation of a curve is \(y = \frac { 1 } { 1 + \tan x }\).

1. Show, by differentiation, that the gradient of the curve is always negative.
2. Use the trapezium rule with 2 intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { 1 + \tan x } \mathrm {~d} x$$ giving your answer correct to 2 significant figures.
3. \includegraphics[max width=\textwidth, alt={}, center]{a31a4b4e-83a6-47d9-9679-3471b3da1b6e-3_556_802_1384_708} The diagram shows a sketch of the curve for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii).

7

1. Differentiate \(\ln ( 2 x + 3 )\).
2. Hence, or otherwise, show that $$\int _ { - 1 } ^ { 3 } \frac { 1 } { 2 x + 3 } \mathrm {~d} x = \ln 3$$
3. Find the quotient and remainder when \(4 x ^ { 2 } + 8 x\) is divided by \(2 x + 3\).
4. Hence show that $$\int _ { - 1 } ^ { 3 } \frac { 4 x ^ { 2 } + 8 x } { 2 x + 3 } d x = 12 - 3 \ln 3$$

6

1. Find \(\int 4 \mathrm { e } ^ { x } \left( 3 + \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
2. Show that \(\int _ { - \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 4 } \pi } \left( 3 + 2 \tan ^ { 2 } \theta \right) \mathrm { d } \theta = \frac { 1 } { 2 } ( 8 + \pi )\).

4

1. Find the value of \(\int _ { 0 } ^ { \frac { 2 } { 3 } \pi } \sin \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
2. Find \(\int \mathrm { e } ^ { - x } \left( 1 + \mathrm { e } ^ { x } \right) \mathrm { d } x\).

5 \includegraphics[max width=\textwidth, alt={}, center]{beb8df77-e091-4248-812b-20e885c42e37-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

5 \includegraphics[max width=\textwidth, alt={}, center]{0a45a806-007f-4840-85e7-16d4c1a2c599-3_528_757_251_694} The diagram shows the curve \(y = 4 e ^ { \frac { 1 } { 2 } x } - 6 x + 3\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) can be written in the form \(\ln a\), where the value of \(a\) is to be stated.
2. Find the exact value of the area of the region enclosed by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { 7 - 2 x }\). The point \(( 3,2 )\) lies on the curve. Find the equation of the curve.

6

1. Show that \(\int _ { 6 } ^ { 16 } \frac { 6 } { 2 x - 7 } \mathrm {~d} x = \ln 125\).
2. Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 17 } \log _ { 10 } x d x$$ giving your answer correct to 3 significant figures.

7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 1 } { 2 } \mathrm { e } ^ { 3 x } + x ^ { 2 } \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 3 } \ln \left( 61 - 2 a ^ { 3 } \right)\).
2. Show by calculation that the value of \(a\) lies between 1.0 and 1.5.
3. Use an iterative formula, based on the equation in part (i), to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

5

1. Given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 1 \right) \mathrm { d } x = 10\), show that the positive constant \(a\) satisfies the equation $$a = 2 \ln \left( \frac { 16 - a } { 6 } \right)$$
2. Use the iterative formula \(a _ { n + 1 } = 2 \ln \left( \frac { 16 - a _ { n } } { 6 } \right)\) with \(a _ { 1 } = 2\) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-2_524_625_1425_758} The diagram shows the curve \(y = \mathrm { e } ^ { x } + 4 \mathrm { e } ^ { - 2 x }\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\).
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0 , x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac { 5 } { 2 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\).
2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

7

1. Find \(\int \frac { 1 + \cos ^ { 4 } 2 x } { \cos ^ { 2 } 2 x } \mathrm {~d} x\).
2. Without using a calculator, find the exact value of \(\int _ { 4 } ^ { 14 } \left( 2 + \frac { 6 } { 3 x - 2 } \right) \mathrm { d } x\), giving your answer in the form \(\ln \left( a \mathrm { e } ^ { b } \right)\), where \(a\) and \(b\) are integers.

6

1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
3. \includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.

6

1. Find \(\int \frac { 4 + \mathrm { e } ^ { x } } { 2 \mathrm { e } ^ { 2 x } } \mathrm {~d} x\).
2. Without using a calculator, find \(\int _ { 2 } ^ { 10 } \frac { 1 } { 2 x + 5 } \mathrm {~d} x\), giving your answer in the form \(\ln k\).
3. \includegraphics[max width=\textwidth, alt={}, center]{f85c4010-17b1-441c-ae8a-e77573d1b0c3-3_446_755_580_735} The diagram shows the curve \(y = \log _ { 10 } ( x + 2 )\) for \(0 \leqslant x \leqslant 6\). The region bounded by the curve and the lines \(x = 0 , x = 6\) and \(y = 0\) is denoted by \(R\). Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 1 decimal place.

3 Given that \(\int _ { 0 } ^ { a } 4 \mathrm { e } ^ { \frac { 1 } { 2 } x + 3 } \mathrm {~d} x = 835\), find the value of the constant \(a\) correct to 3 significant figures. [5]

6 \includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-06_561_542_260_799} The diagram shows the curve \(y = \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\). The shaded region is bounded by the curve and the lines \(x = \frac { 1 } { 6 } \pi\) and \(y = 0\).

1. Use the trapezium rule with two intervals to find an approximation to the area of the shaded region, giving your answer correct to 3 significant figures.
2. Find the exact volume of the solid formed when the shaded region is rotated completely about the \(x\)-axis.

7

1. Find \(\int ( 2 \cos \theta - 3 ) ( \cos \theta + 1 ) \mathrm { d } \theta\).
2. 1. Find \(\int \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x\).
   2. Hence find \(\int _ { 1 } ^ { 4 } \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x\), giving your answer in the form \(\ln k\).

3 Without using a calculator, find the exact value of \(\int _ { 0 } ^ { 2 } 4 \mathrm { e } ^ { - x } \left( \mathrm { e } ^ { 3 x } + 1 \right) \mathrm { d } x\).

6 It is given that \(\int _ { 0 } ^ { a } \left( 1 + \mathrm { e } ^ { \frac { 1 } { 2 } x } \right) ^ { 2 } \mathrm {~d} x = 10\), where \(a\) is a positive constant.

1. Show that \(a = 2 \ln \left( \frac { 15 - a } { 4 + \mathrm { e } ^ { \frac { 1 } { 2 } a } } \right)\).
2. Use the equation in part (i) to show by calculation that \(1.5 < a < 1.6\).
3. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.