Find constant from definite integral

10 A curve has equation \(y = \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } }\) where \(x > 0\) and \(k\) is a positive constant.

1. It is given that when \(x = \frac { 1 } { 4 }\), the gradient of the curve is 3 . Find the value of \(k\).
2. It is given instead that \(\int _ { \frac { 1 } { 4 } k ^ { 2 } } ^ { k ^ { 2 } } \left( \frac { 1 } { k } x ^ { \frac { 1 } { 2 } } + x ^ { - \frac { 1 } { 2 } } + \frac { 1 } { k ^ { 2 } } \right) \mathrm { d } x = \frac { 13 } { 12 }\). Find the value of \(k\).

2 A curve with equation \(y = \mathrm { f } ( x )\) passes through the points \(( 0,2 )\) and \(( 3 , - 1 )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = k x ^ { 2 } - 2 x\), where \(k\) is a constant. Find the value of \(k\).

8 A curve passes through \(( 0,11 )\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants.

1. Find the equation of the curve in terms of \(a\) and \(b\).
2. It is now given that the curve has a stationary point at \(( 2,3 )\). Find the values of \(a\) and \(b\).

7 It is given that \(\int _ { 0 } ^ { a } \left( \frac { 4 } { 2 x + 1 } + 8 x \right) \mathrm { d } x = 10\), where \(a\) is a positive constant.

1. Show that \(a = \sqrt { 2.5 - 0.5 \ln ( 2 a + 1 ) }\).
2. Using the equation in part (a), show by calculation that \(1 < a < 2\).
3. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

3 It is given that \(\int _ { a } ^ { 3 a } \frac { 2 } { 2 x - 5 } \mathrm {~d} x = \ln \frac { 7 } { 2 }\).
Find the value of the positive constant \(a\).

6

1. Find \(\int \left( \frac { 8 } { 4 x + 1 } + \frac { 8 } { \cos ^ { 2 } ( 4 x + 1 ) } \right) \mathrm { d } x\).
2. It is given that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 3 + 4 \cos ^ { 2 } \frac { 1 } { 2 } x + k \sin 2 x \right) \mathrm { d } x = 10\). Find the exact value of the constant \(k\).

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

5 It is given that \(\int _ { a } ^ { a ^ { 3 } } \frac { 10 } { 2 x + 1 } \mathrm {~d} x = 7\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }\).
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2. Use an iterative formula, based on the equation in part (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 2 and give the result of each iteration to 5 significant figures.

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]

1 Find the exact value of the constant \(k\) for which \(\int _ { 1 } ^ { k } \frac { 1 } { 2 x - 1 } \mathrm {~d} x = 1\).

5 Given that \(\int _ { 0 } ^ { a } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x = 65\), find the value of \(a\) correct to 3 decimal places.

4 Find the exact value of the positive constant \(k\) for which $$\int _ { 0 } ^ { k } e ^ { 4 x } d x = \int _ { 0 } ^ { 2 k } e ^ { x } d x$$

6

1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin x ( 4 \sin x + 6 \cos x ) \mathrm { d } x\).
2. Given that \(\int _ { 0 } ^ { a } \frac { 6 } { 3 x + 2 } \mathrm {~d} x = \ln 49\), find the value of the positive constant \(a\).

5 It is given that \(\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } + 4 \cos 2 x - \sin x \right) \mathrm { d } x = 2\), where \(a\) is a constant.

1. Show that \(a = \sqrt [ 3 ] { } ( 3 - 2 \sin 2 a - \cos a )\).
2. Using the equation in part (i), show by calculation that \(0.5 < a < 0.75\).
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.

7. (i) A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,3 )\). Given that $$f ^ { \prime } ( x ) = \frac { 4 } { x ^ { 3 } } + 2 x - 1$$ find the value of \(\mathrm { f } ( 1 )\).
(ii) Given that $$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 21$$ find the exact value of the constant \(A\).

6 Given that $$\int _ { 0 } ^ { \ln 4 } \left( k \mathrm { e } ^ { 3 x } + ( k - 2 ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x = 185$$ find the value of the constant \(k\).

1. Leaking oil is forming a circular patch on the surface of the sea. The area of the patch is increasing at a rate of 250 square metres per hour. Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 1900 square metres. Give your answer correct to 2 significant figures.
2. The mass of a substance is decreasing exponentially. Its mass now is 150 grams and its mass, \(m\) grams, at a time \(t\) years from now is given by $$m = 150 \mathrm { e } ^ { - k t } ,$$ where \(k\) is a positive constant. Find, in terms of \(k\), the number of years from now at which the mass will be decreasing at a rate of 3 grams per year.
   1. The curve \(y = \sqrt { x }\) can be transformed to the curve \(y = \sqrt { 2 x + 3 }\) by means of a stretch parallel to the \(y\)-axis followed by a translation. State the scale factor of the stretch and give details of the translation.
   2. It is given that \(N\) is a positive integer. By sketching on a single diagram the graphs of \(y = \sqrt { 2 x + 3 }\) and \(y = \frac { N } { x ^ { 3 } }\), show that the equation $$\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }$$ has exactly one real root.
   3. A sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) has the property that $$x _ { n + 1 } = N ^ { \frac { 1 } { 3 } } \left( 2 x _ { n } + 3 \right) ^ { - \frac { 1 } { 6 } }$$ For certain values of \(x _ { 1 }\) and \(N\), it is given that the sequence converges to the root of the equation \(\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }\).

4 It is given that \(\int _ { a } ^ { 3 a } \left( \mathrm { e } ^ { 3 x } + \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 9 } \ln \left( 300 + 3 \mathrm { e } ^ { a } - 2 \mathrm { e } ^ { 3 a } \right)\).
2. Use an iterative process, based on the equation in part (i), to find the value of \(a\) correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process.

1. (a) Given that \(k\) is a constant, find

$$\int \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x$$ simplifying your answer.
(b) Hence find the value of \(k\) such that $$\int _ { 0.5 } ^ { 2 } \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x = 8$$