Integration by Substitution

1. Use the substitution \(u = 2 - x^2\) to find $$\int \frac{x}{2 - x^2} \, dx.$$ [4]
2. Evaluate $$\int_0^{\frac{1}{4}} \sin 3x \cos x \, dx.$$ [6]

10

1. Show that \(\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln \left( \frac { 3 } { \sqrt { 5 } } \right)\).
2. Find \(\int x \sqrt { x - 2 } \mathrm {~d} x\).

8

1. The curve \(C _ { 1 }\) has equation \(y = - \ln ( \cos x )\). Show that the length of the arc of \(C _ { 1 }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 3 } \pi\) is \(\ln ( 2 + \sqrt { 3 } )\).
2. The curve \(C _ { 2 }\) has equation \(y = 2 \sqrt { } ( x + 3 )\). The arc of \(C _ { 2 }\) joining the point where \(x = 0\) to the point where \(x = 1\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface generated is $$\frac { 8 } { 3 } \pi ( 5 \sqrt { } 5 - 8 )$$

Show that \(\int_1^4 \frac{x}{x^2 + 2} \, dx = \frac{1}{2} \ln 6\). [4]

Use the substitution \(u = 2x - 5\) to show that \(\int_2^3 (4x - 8)(2x - 5)^7 \, dx = \frac{17}{72}\). [5]

10

1. Use the substitution \(u = \tan x\) to show that, for \(n \neq - 1\), $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
2. Hence find the exact value of
   1. \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x\),
   2. \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 2 x + 5 )\) and \(( 2,5 )\) is a point on the curve. Find the equation of the curve.

3 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 ( 4 x - 7 ) ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(\left( 4 , \frac { 5 } { 2 } \right)\). Find the equation of the curve.

1. (a) Use the substitution \(u = 4 - \sqrt { h }\) to show that

$$\int \frac { \mathrm { d } h } { 4 - \sqrt { h } } = - 8 \ln | 4 - \sqrt { h } | - 2 \sqrt { h } + k$$ where \(k\) is a constant A team of scientists is studying a species of slow growing tree.
The rate of change in height of a tree in this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { t ^ { 0.25 } ( 4 - \sqrt { h } ) } { 20 }$$ where \(h\) is the height in metres and \(t\) is the time, measured in years, after the tree is planted.
(b) Find, according to the model, the range in heights of trees in this species. One of these trees is one metre high when it is first planted.
According to the model,
(c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures.

Evaluate $$\int_2^6 \sqrt{3x-2} \, dx.$$ [4]

4 Evaluate the following integrals, giving your answers in exact form.

1. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x ^ { 2 } + 1 } \mathrm {~d} x\).
2. \(\int _ { 0 } ^ { 1 } \frac { 2 x } { x + 1 } \mathrm {~d} x\).

1. (i) Use the substitution \(t = \tan \frac { X } { 2 }\) to prove the identity

$$\frac { \sin x - \cos x + 1 } { \sin x + \cos x - 1 } \equiv \sec x + \tan x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$ (ii) Use the substitution \(t = \tan \frac { \theta } { 2 }\) to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { 5 } { 4 + 2 \cos \theta } d \theta$$ giving your answer in simplest form.

Evaluate \(\int_0^3 x(x + 1)^{-\frac{1}{2}} dx\), giving your answer as an exact fraction. [5]

6 Use the substitution \(u = 2 x + 1\) to evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 4 x - 1 } { ( 2 x + 1 ) ^ { 5 } } \mathrm {~d} x\).

3. Using the substitution \(u = 2 + \sqrt { } ( 2 x + 1 )\), or other suitable substitutions, find the exact value of $$\int _ { 0 } ^ { 4 } \frac { 1 } { 2 + \sqrt { } ( 2 x + 1 ) } d x$$ giving your answer in the form \(A + 2 \ln B\), where \(A\) is an integer and \(B\) is a positive constant.

Use the substitution \(x = \sin \theta\) to find the exact value of $$\int_0^1 \frac{1}{(1-x^2)^{3/2}} dx.$$ [7]

4. Use the substitution \(x = \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$

1. Using a suitable substitution, find $$\int \sqrt{1 - x^2} \, dx.$$ [4]
2. Show that the differential equation $$\frac{dy}{dx} = 2\sqrt{1 - x^2 - y^2 + x^2y^2},$$ given that \(y = 0\) when \(x = 0\), \(|x| < 1\) and \(|y| < 1\), has the solution $$y = x \cos\left(x\sqrt{1 - x^2}\right) + \sqrt{1 - x^2} \sin\left(x\sqrt{1 - x^2}\right).$$ [5]

2 Use the substitution \(u = 3 x + 1\) to find \(\int \frac { 3 x } { 3 x + 1 } \mathrm {~d} x\).

5. (i) Find $$\int \left( ( 3 x + 5 ) ^ { 9 } + \mathrm { e } ^ { 5 x } \right) \mathrm { d } x$$ (ii) Given that \(b\) is a constant greater than 2 , and $$\int _ { 2 } ^ { b } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln ( \sqrt { 6 } )$$ use integration to find the value of \(b\).

6

1. Given that \(y = ( x - 2 ) \sqrt { 5 + 4 x - x ^ { 2 } } + 9 \sin ^ { - 1 } \left( \frac { x - 2 } { 3 } \right)\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 5 + 4 x - x ^ { 2 } }$$ where \(k\) is an integer.
2. Hence show that $$\int _ { 2 } ^ { \frac { 7 } { 2 } } \sqrt { 5 + 4 x - x ^ { 2 } } \mathrm {~d} x = p \sqrt { 3 } + q \pi$$ where \(p\) and \(q\) are rational numbers.
   [0pt] [3 marks]

1 Find \(\int \sqrt [ 3 ] { 2 x - 1 } \mathrm {~d} x\).

1. (i) Find

$$\int \frac { 12 } { ( 2 x - 1 ) ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) (a) Write \(\frac { 4 x + 3 } { x + 2 }\) in the form $$A + \frac { B } { x + 2 } \text { where } A \text { and } B \text { are constants to be found }$$ (b) Hence find, using algebraic integration, the exact value of $$\int _ { - 8 } ^ { - 5 } \frac { 4 x + 3 } { x + 2 } d x$$ giving your answer in simplest form.

8. (i) Find, using algebraic integration, the exact value of $$\int _ { 3 } ^ { 42 } \frac { 2 } { 3 x - 1 } \mathrm {~d} x$$ giving your answer in simplest form.
(ii) $$\mathrm { h } ( x ) = \frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 1 } { ( x - 1 ) ^ { 2 } } \quad x > 1$$ Given \(\mathrm { h } ( x ) = A x + B + \frac { C } { ( x - 1 ) ^ { 2 } }\) where \(A , B\) and \(C\) are constants to be found, find $$\int \mathrm { h } ( x ) \mathrm { d } x$$ \includegraphics[max width=\textwidth, alt={}, center]{1c700103-ecab-4a08-b411-3f445ed88885-26_2258_47_312_1985}

Evaluate \(\int_0^1 \frac{1}{1 + \sqrt{x}} \, dx\), giving your answer in the form \(a + b \ln c\), where \(a\), \(b\) and \(c\) are integers. [6]

4 Use the substitution \(u = \sqrt { x + 2 }\) to find the exact value of $$\int _ { - 1 } ^ { 7 } \frac { x ^ { 2 } } { \sqrt { x + 2 } } \mathrm {~d} x$$

1. Given that \(u = 2^x\), write down an expression for \(\frac{du}{dx}\) [1 mark]
2. Find the exact value of \(\int_0^1 2^x \sqrt{3 + 2^x}\) dx Fully justify your answer. [6 marks]

1. Find the exact coordinates of \(M\).
2. Using the substitution \(u = 3 - 2 x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt { 13 }\), where \(a\) is a rational number. [5]

9 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_849_565_1466_790} The diagram shows part of the curve \(y = 8 - \sqrt { } ( 4 - x )\) and the tangent to the curve at \(P ( 3,7 )\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\).
3. Find, showing all necessary working, the area of the shaded region.

10

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin 3 x - 3 x \cos 3 x ) = 9 x \sin 3 x\). The curve shown in the figure below is part of the graph of the function \(y = x \sin 3 x\). \includegraphics[max width=\textwidth, alt={}, center]{3e4281d1-dbad-46a2-bbb7-97706bda2dfa-3_508_1136_1939_466}
2. Show that \(\int _ { 0 } ^ { \frac { 2 \pi } { 3 } } | x \sin 3 x | \mathrm { d } x = \frac { 4 \pi } { 9 }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{208eab3e-a78c-43b4-918f-a9efc9b4f47a-4_429_748_264_699} The diagram shows part of the curve \(y = \frac { x } { x ^ { 2 } + 1 }\) and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and by the lines \(y = 0\) and \(x = p\).

1. Calculate the \(x\)-coordinate of \(M\).
2. Find the area of \(R\) in terms of \(p\).
3. Hence calculate the value of \(p\) for which the area of \(R\) is 1 , giving your answer correct to 3 significant figures.

11 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve \(y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }\) and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
   The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.
3. Find, showing all necessary working, the 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.

10 \includegraphics[max width=\textwidth, alt={}, center]{0f73e750-18a0-49ad-b4cb-fd6d14f0789e-4_424_713_262_715} The diagram shows the curve \(y = x ^ { 2 } \sqrt { } \left( 1 - x ^ { 2 } \right)\) for \(x \geqslant 0\) and its maximum point \(M\).

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Show, by means of the substitution \(x = \sin \theta\), that the area \(A\) of the shaded region between the curve and the \(x\)-axis is given by $$A = \frac { 1 } { 4 } \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$
3. Hence obtain the exact value of \(A\).

Using partial fractions, find \(\int \frac{x}{(x+1)(2x+1)} dx\). [7]

Use the substitution \(u = 1 - x^2\) to find $$\int \frac{1}{1-x^2} \, dx.$$ [6]

5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 3 x - 2 ) }\). Given that the curve passes through the point \(P ( 2,11 )\), find

1. the equation of the normal to the curve at \(P\),
2. the equation of the curve.

10 By first using the substitution \(u = \mathrm { e } ^ { x }\), show that $$\int _ { 0 } ^ { \ln 4 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } + 3 \mathrm { e } ^ { x } + 2 } \mathrm {~d} x = \ln \left( \frac { 8 } { 5 } \right)$$

6

1. Show that the substitution \(x = \sin ^ { 2 } \theta\) transforms \(\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\) to \(\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta\).
2. Hence find \(\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x\).

\includegraphics{figure_6} Figure 1 shows the curve with equation \(y = x\sqrt{1-x}\), \(0 \leq x \leq 1\).

1. Use the substitution \(u^2 = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac{8}{15}\). [8]
2. Find, in terms of \(\pi\), the volume of the solid formed when the region bounded by the curve and the \(x\)-axis is rotated through \(360°\) about the \(x\)-axis. [5]

16 In this question you must show detailed reasoning.
Find \(\int \frac { x } { 1 + \sqrt { x } } d x\). END OF QUESTION PAPER

7

1. Find \(\int 5 x ^ { 3 } \sqrt { x ^ { 2 } + 1 } \mathrm {~d} x\).
2. Find \(\int \theta \tan ^ { 2 } \theta \mathrm {~d} \theta\). You may use the result \(\int \tan \theta \mathrm { d } \theta = \ln | \sec \theta | + c\).

Use the substitution \(x = 2\tan u\) to show that $$\int_0^2 \frac{x^2}{x^2 + 4} \, dx = \frac{1}{2}(4 - \pi).$$ [8]

9. a. Use the substitution \(t ^ { 2 } = 2 x - 5\) to show that $$\int \frac { 1 } { x + 3 \sqrt { 2 x - 5 } } \mathrm {~d} x = \int \frac { 2 t } { t ^ { 2 } + 6 t + 5 } \mathrm {~d} t$$ b. Hence find the exact value of $$\int _ { 3 } ^ { 27 } \frac { 1 } { x + 3 \sqrt { 2 x - 5 } } \mathrm {~d} x$$

Show that $$\int_0^{\pi/2} \frac{\sin 2\theta}{1 + \cos \theta} \, d\theta = 2 - 2\ln 2$$ [7]

7 In this question you must show detailed reasoning.
Use the substitution \(u = 6 x ^ { 2 } + x\) to solve the equation \(36 x ^ { 4 } + 12 x ^ { 3 } + 7 x ^ { 2 } + x - 2 = 0\).

1. Show that the substitution \(x = \tan \theta\) transforms \(\int \frac{1}{(1 + x^2)^2} dx\) to \(\int \cos^2 \theta d\theta\). [3]
2. Hence find the exact value of \(\int_0^1 \frac{1}{(1 + x^2)^2} dx\). [4]

2 Use the substitution \(u = 1 + 3 \tan x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { } ( 1 + 3 \tan x ) } { \cos ^ { 2 } x } d x$$

6 Let \(I = \int _ { 0 } ^ { 1 } \frac { \sqrt { } x } { 2 - \sqrt { } x } \mathrm {~d} x\).

1. Using the substitution \(u = 2 - \sqrt { } x\), show that \(I = \int _ { 1 } ^ { 2 } \frac { 2 ( 2 - u ) ^ { 2 } } { u } \mathrm {~d} u\).
2. Hence show that \(I = 8 \ln 2 - 5\).

5 By first using the substitution \(t = \sqrt { x + 1 }\), find \(\int \mathrm { e } ^ { 2 \sqrt { x + 1 } } \mathrm {~d} x\).

1. Using the substitution \(u = x^2\), or otherwise, find the numerical value of $$\int_0^{\sqrt{\ln 4}} xe^{-\frac{1}{2}x^2} \, dx.$$ [4]
2. Determine the exact coordinates of the stationary points of the curve \(y = xe^{-\frac{1}{2}x^2}\). [4]

5 Use the substitution \(u = \cos x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$

1 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to find \(\int \frac { 1 } { 1 + \sin x + \cos x } \mathrm {~d} x\).

4 Using the substitution \(u = \sqrt { x }\), find the exact value of $$\int _ { 3 } ^ { \infty } \frac { 1 } { ( x + 1 ) \sqrt { x } } \mathrm {~d} x$$

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 4 x + 1 )\) and \(( 2,5 )\) is a point on the curve.

1. Find the equation of the curve.
2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \(( 2,5 )\).
3. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \times \frac { \mathrm { d } y } { \mathrm {~d} x }\) is constant.

5 Use the substitution \(u = \cos x\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \cos ^ { 2 } x d x$$

\includegraphics{figure_7} The diagram shows the curve \(y = 5\sin^2 x \cos^3 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places. [5]
2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]