Integration using inverse trig and hyperbolic functions

11

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 2 } \tan ^ { - 1 } x \right)\) 11
2. Hence find \(\int 2 x \tan ^ { - 1 } x \mathrm {~d} x\)

11

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 2 } \tan ^ { - 1 } x \right)\) 11
2. Hence find \(\int 2 x \tan ^ { - 1 } x \mathrm {~d} x\)

5. Given that \(y = \operatorname { artanh } ( \cos x )\)

1. show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \cos x \operatorname { artanh } ( \cos x ) d x$$ giving your answer in the form \(a \ln ( b + c \sqrt { 3 } ) + d \pi\), where \(a , b , c\) and \(d\) are rational numbers to be found.
   (5)
   

1 Find \(\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + x ^ { 2 } } } \mathrm {~d} x\), giving your answer exactly in logarithmic form.

2. (a) Find $$\int \frac { 1 } { \sqrt { } \left( 4 x ^ { 2 } + 9 \right) } d x$$ (b) Use your answer to part (a) to find the exact value of $$\int _ { - 3 } ^ { 3 } \frac { 1 } { \sqrt { \left( 4 x ^ { 2 } + 9 \right) } } d x$$ giving your answer in the form \(k \ln ( a + b \sqrt { } 5 )\), where \(a\) and \(b\) are integers and \(k\) is a constant.

1. The curve \(C\) has equation

$$y = \frac { 1 } { \sqrt { x ^ { 2 } + 2 x - 3 } } , \quad x > 1$$

1. Find \(\int y \mathrm {~d} x\) The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 3\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the volume of the solid generated. Give your answer in the form \(p \pi \ln q\), where \(p\) and \(q\) are rational numbers to be found.

1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).

2. Determine

1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)

1 Find the exact value of \(\int _ { 2 } ^ { \frac { 7 } { 2 } } \frac { 1 } { \sqrt { 4 x - x ^ { 2 } - 1 } } \mathrm {~d} x\).

16 The point \(P ( 4,1,0 )\) is equidistant from the plane \(2 x + y + 2 z = 0\) and the line \(\frac { x - 3 } { 2 } = \frac { y - 1 } { b } = \frac { z + 5 } { 3 }\), where \(b > 0\). Determine the value of \(b\).

5 Let \(I = \int _ { 0 } ^ { 1 } \frac { 9 } { \left( 3 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x\).

1. Using the substitution \(x = ( \sqrt { } 3 ) \tan \theta\), show that \(I = \sqrt { } 3 \int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \cos ^ { 2 } \theta \mathrm {~d} \theta\).
2. Hence find the exact value of \(I\).

5. $$I = \int \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x , \quad x > 1$$ （a）Use the substitution \(x = 1 + u ^ { - 1 }\) to show that $$I = - \left( \frac { x + 1 } { x - 1 } \right) ^ { \frac { 1 } { 2 } } + c$$ （b）Hence show that $$\int _ { \sec \alpha } ^ { \sec \beta } \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x = \cot \left( \frac { \alpha } { 2 } \right) - \cot \left( \frac { \beta } { 2 } \right) , \quad 0 < \alpha < \beta < \frac { \pi } { 2 }$$

2 Fid \(\mathbf { b }\) ex ct le \(\mathbf { 6 } \int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 3 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\).

14 The function f is defined by $$f(x) = \frac{1}{4x^2 + 16x + 19} \quad (x \in \mathbb{R})$$ 14

1. Show, without using calculus, that the graph of \(y = f(x)\) has a stationary point at \(\left(-2, \frac{1}{3}\right)\)
   [3 marks] 14
2. Show that \(\int_{-2}^{-\frac{1}{2}} f(x) \, dx = \frac{\pi\sqrt{3}}{18}\)
   [5 marks] 14
3. Find the value of \(\int_{-2}^{\infty} f(x) \, dx\)
   Fully justify your answer.
   [2 marks]

2. Use calculus to find the exact value of \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\).

1 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } 2 x\) and \(\mathrm { g } ( x ) = p \tan ^ { - 1 } x\), where \(p\) is a constant. Find the value of \(p\) for which \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 2 } \right) = \mathrm { g } ^ { \prime } \left( \frac { 1 } { 2 } \right)\).

11. (a) Prove that the derivative of \(\operatorname { artanh } x , - 1 < x < 1\), is \(\frac { 1 } { 1 - x ^ { 2 } }\).
(3)
(b) Find \(\int \operatorname { artanh } x \mathrm {~d} x\).
(4)
[0pt] [P5 June 2003 Qn 2]

3 Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 - \sin x } \mathrm {~d} x = 1 + \sqrt { 3 }$$

17 By making a suitable substitution, show that $$\int _ { - 2 } ^ { 1 } \sqrt { x ^ { 2 } + 6 x + 8 } d x = 2 \sqrt { 15 } - \frac { 1 } { 2 } \cosh ^ { - 1 } ( 4 )$$

14. Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
[0pt] [BLANK PAGE]

4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.

6 [In this question you may use, without proof, the formula \(\int \sec x \mathrm {~d} x = \ln ( \sec x + \tan x ) + \operatorname { const }\).]

1. Let \(y = \sec x\). Find the mean value of \(y\) with respect to \(x\) over the interval \(\frac { 1 } { 6 } \pi \leqslant x \leqslant \frac { 1 } { 3 } \pi\).
2. The curve \(C\) has equation \(y = - \ln ( \cos x )\), for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). Find the arc length of \(C\).

1 Find the value of \(\int _ { 6 } ^ { 7 } \frac { 1 } { \sqrt { ( x - 5 ) ^ { 2 } - 1 } } \mathrm {~d} x\), giving your answer in the form \(\ln ( a + \sqrt { b } )\), where \(a\) and \(b\) are integers to be determined.

7

1. Given that \(y = \tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right)\) and \(x \neq 1\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
   [0pt] [4 marks]
2. Hence, given that \(x < 1\), show that \(\tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right) - \tan ^ { - 1 } x = \frac { \pi } { 4 }\).
   [0pt] [3 marks]