Show substitution transforms integral, then apply integration by parts or further substitution

1. (a) Use the substitution \(x = \sec \theta\) to show that

$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$ can be written as $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$ (3)
(b) Use integration by parts to show that $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$ (c) Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x\).

5 Show that \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } - \frac { 1 } { 2 \mathrm { e } }\). Let \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x\). Show that \(I _ { 2 n + 1 } = n I _ { 2 n - 1 } - \frac { 1 } { 2 \mathrm { e } }\) for \(n \geqslant 1\). Find the exact value of \(I _ { 7 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-7_352_545_258_239} The diagram shows the curve \(y = \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right)\), for \(1 \leqslant x \leqslant 2\).

1. Use rectangles of width 0.25 to find upper and lower bounds for \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x\). Give your answers correct to 3 significant figures.
2. 1. Use the substitution \(t = \sqrt { x - 1 }\) to show that \(\int \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = \int 2 t \sin \left( \frac { 1 } { 2 } t \right) \mathrm { d } t\).
   2. Hence show that \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = 8 \sin \frac { 1 } { 2 } - 4 \cos \frac { 1 } { 2 }\).

4.(a)Use the substitution \(x = \sqrt { 3 } \tan u\) to show that $$\int \frac { 1 } { 3 + x ^ { 2 } } \mathrm {~d} x = p \arctan ( p x ) + c$$ where \(p\) is a real constant to be determined and \(c\) is an arbitrary constant.
(b)Use the substitution \(x = \frac { 3 u + 3 } { u - 3 }\) to determine the exact value of \(I\) where $$I = \int _ { - 3 } ^ { 1 } \frac { \ln ( 3 - x ) } { 3 + x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-10_2264_47_314_1984}

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 \(\int \tan \theta \mathrm { d } \theta = \ln | \sec \theta | + c\).

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Use the substitution \(u = e^x - 3\) to show that $$\int_{\ln 5}^{\ln 7} \frac{4e^{3x}}{e^x - 3} \, dx = a + b \ln 2$$ where \(a\) and \(b\) are constants to be found. [7]
2. Show, by integration, that $$\int 3e^x \cos 2x \, dx = pe^x \sin 2x + qe^x \cos 2x + c$$ where \(p\) and \(q\) are constants to be found and \(c\) is an arbitrary constant. [5]