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 \(k\).

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 \(a = \sqrt [ 3 ] { 0.5 \mathrm { e } ^ { 1.4 } ( 2 a + 1 ) - 0.5 }\). \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-08_2718_35_107_2011} \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-09_2725_35_99_20}
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 \(A\).

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 $$\int _ { 0.5 } ^ { 2 } \left( \frac { 4 } { x ^ { 3 } } + k x \right) \mathrm { d } x = 8$$

Given that $$\int_0^{\ln 4} (ke^{3x} + (k - 2)e^{-\frac{x}{3}}) \, dx = 185,$$ find the value of the constant \(k\). [7]