4.08h Integration: inverse trig/hyperbolic substitutions

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

6 By first completing the square, find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x ^ { 2 } + 4 x + 8 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.

6 Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of \(a\).

6 You are given that \(y = \tan ^ { - 1 } \sqrt { 2 x }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 2 } } \frac { \sqrt { x } } { \left( x + 2 x ^ { 2 } \right) } \mathrm { d } x = k \pi\) where \(k\) is a number to be determined in exact form.

7 The function sech \(x\) is defined by \(\operatorname { sech } x = \frac { 1 } { \cosh x }\).

1. Show that \(\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }\).
2. Using a suitable substitution, find \(\int \operatorname { sech } x \mathrm {~d} x\).

8

1. Using exponentials, show that \(\cosh 2 u \equiv 2 \sinh ^ { 2 } u + 1\).
2. By differentiating both sides of the identity in part (a) with respect to \(u\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\).
3. Use the substitution \(\mathrm { x } = \sinh ^ { 2 } \mathrm { u }\) to find \(\int \sqrt { \frac { x } { x + 1 } } \mathrm {~d} x\). Give your answer in the form asinh \(^ { - 1 } \mathrm {~b} \sqrt { \mathrm { x } } + \mathrm { f } ( \mathrm { x } )\) where \(a\) and \(b\) are integers and \(\mathrm { f } ( x )\) is a function to be determined.
4. Hence determine the exact area of the region between the curve \(\mathrm { y } = \sqrt { \frac { \mathrm { x } } { \mathrm { x } + 1 } }\), the \(x\)-axis, the line \(x = 1\) and the line \(x = 2\). Give your answer in the form \(\mathrm { p } + \mathrm { q } \mid \mathrm { nr }\) where \(p , q\) and \(r\) are numbers to be determined.

12 Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } d x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.

5

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }\).
2. Find \(\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x\).

6

1. Show that the substitution \(t = \tan \theta\) transforms the integral $$\int \frac { \mathrm { d } \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$$ into $$\int \frac { \mathrm { d } t } { 9 + t ^ { 2 } }$$
2. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { d \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } = \frac { \pi } { 18 }$$

6

1. Express \(7 + 4 x - 2 x ^ { 2 }\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are integers.
2. By means of a suitable substitution, or otherwise, find the exact value of $$\int _ { 1 } ^ { \frac { 5 } { 2 } } \frac { \mathrm {~d} x } { \sqrt { 7 + 4 x - 2 x ^ { 2 } } }$$

13

1. Using the logarithmic form of \(\operatorname { arcosh } x\), prove that the derivative of \(\operatorname { arcosh } x\) is \(\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
2. Hence find \(\int _ { 1 } ^ { 2 } \operatorname { arcosh } x \mathrm {~d} x\), giving your answer in exact logarithmic form.
3. Ali tries to evaluate \(\int _ { 0 } ^ { 1 } \operatorname { arcosh } x \mathrm {~d} x\) using his calculator, and gets an 'error'. Explain why.

3 In this question you must show detailed reasoning.
Find \(\int _ { 0 } ^ { \frac { 1 } { 3 } } \frac { 1 } { \sqrt { 4 - 9 x ^ { 2 } } } \mathrm {~d} x\), expressing your answer in terms of \(\pi\).

16

1. Show using exponentials that \(\cosh 2 u = 1 + 2 \sinh ^ { 2 } u\).
2. Show that \(\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x = 2 \sqrt { 2 } - 2 \ln ( 1 + \sqrt { 2 } )\).

3. $$f ( x ) = \frac { 1 } { \sqrt { 4 x ^ { 2 } + 9 } }$$

1. Using a substitution, that should be stated clearly, show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A \sinh ^ { - 1 } ( B x ) + c$$ where \(c\) is an arbitrary constant and \(A\) and \(B\) are constants to be found.
2. Hence find, in exact form in terms of natural logarithms, the mean value of \(\mathrm { f } ( x )\) over the interval \([ 0,3 ]\).

8. $$f ( x ) = \frac { 3 } { 13 + 6 \sin x - 5 \cos x }$$ Using the substitution \(t = \tan \left( \frac { x } { 2 } \right)\)

1. show that \(\mathrm { f } ( x )\) can be written in the form $$\frac { 3 \left( 1 + t ^ { 2 } \right) } { 2 ( 3 t + 1 ) ^ { 2 } + 6 }$$
2. Hence solve, for \(0 < x < 2 \pi\), the equation $$\mathrm { f } ( x ) = \frac { 3 } { 7 }$$ giving your answers to 2 decimal places where appropriate.
3. Use the result of part (a) to show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { 4 \pi } { 3 } } f ( x ) d x = K \left( \arctan \left( \frac { \sqrt { 3 } - 9 } { 3 } \right) - \arctan \left( \frac { \sqrt { 3 } + 3 } { 3 } \right) + \pi \right)$$ where \(K\) is a constant to be determined.

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.

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

1. Use the substitution \(t = \tan \left( \frac { \theta } { 2 } \right)\) to show that $$\int \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \int \frac { a } { ( t + b ) ^ { 2 } + c } d t$$ where \(a\), \(b\) and \(c\) are constants to be determined.
2. Hence show that $$\int _ { \frac { \pi } { 2 } } ^ { \frac { 2 \pi } { 3 } } \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \ln \left( \frac { 2 \sqrt { 3 } } { 3 } \right)$$

15 In this question you must show detailed reasoning. Show that \(\int _ { \frac { 3 } { 4 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 4 x + 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { 3 + \sqrt { 5 } } { 2 } \right)\).

15 In this question you must show detailed reasoning. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 1 + 2 x - x ^ { 2 } } } d x\), giving your answer in terms of \(\pi\).

16 In this question you must show detailed reasoning. Show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } } \mathrm { dx } = \ln \left( \frac { \mathrm { a } + \mathrm { b } \sqrt { 3 } } { \mathrm { c } } \right)\), where \(a , b\) and \(c\) are integers to be determined.

4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm . \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$

4 In this question you must show detailed reasoning.
Find the exact value of each of the following.

1. \(\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x + 10 } \mathrm {~d} x\)
2. The mean value of \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) in the interval \([ 0,0.5 ]\)

9 In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x\) giving your answer in the form \(\ln ( p + q \sqrt { 2 } )\) where \(p\) and \(q\) are integers to be determined.

7 A curve has equation \(y = 4 \sqrt { x }\).

1. Show that the length of arc \(s\) of the curve between the points where \(x = 0\) and \(x = 1\) is given by $$s = \int _ { 0 } ^ { 1 } \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x$$
2. 1. Use the substitution \(x = 4 \sinh ^ { 2 } \theta\) to show that $$\int \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x = \int 8 \cosh ^ { 2 } \theta \mathrm {~d} \theta$$
   2. Hence show that $$s = 4 \sinh ^ { - 1 } 0.5 + \sqrt { 5 }$$