4.08c Improper integrals: infinite limits or discontinuous integrands

1. (i) (a) Explain why \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is an improper integral.
   (b) Show that \(\int _ { 0 } ^ { \infty } \cosh x \mathrm {~d} x\) is divergent.
   (ii)

$$4 \sinh x = p \cosh x \quad \text { where } p \text { is a real constant }$$ Given that this equation has real solutions, determine the range of possible values for \(p\)

1. (a) Explain why

$$\int _ { \frac { 4 } { 3 } } ^ { \infty } \frac { 1 } { 9 x ^ { 2 } + 16 } d x$$ is an improper integral.
(b) Show that $$\int _ { \frac { 4 } { 3 } } ^ { \infty } \frac { 1 } { 9 x ^ { 2 } + 16 } d x = k \pi$$ where \(k\) is a constant to be determined.

9 In this question you must show detailed reasoning.

1. Use de Moivre's theorem to determine constants \(A\), \(B\) and \(C\) such that $$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$ The function f is defined by \(\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1\).
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\). \includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260} The diagram shows the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\) for \(0 \leqslant x < 1\) and the asymptote \(x = 1\). The region \(R\) is the unbounded region between the curve, the \(x\)-axis, the line \(x = 0\) and the line \(x = 1\). You are given that the area of \(R\) is finite.
3. Determine the exact area of \(R\).

5 In this question you must show detailed reasoning. Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]

1. (i) Given that

$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant \(k\).
(ii) Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.

4 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:

1. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x\);
   (3 marks)
2. \(\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x\);
3. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x\).

5

1. Explain why \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\) is an improper integral.
2. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
   1. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\);
   2. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\).

8

1. The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
   1. Express \(\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
   2. Use your answer to part (a)(i) to find the value of \(f ^ { \prime } ( 1 )\).
2. The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
   The graphs intersect at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\mathrm { f } ( x ) = 0\), where f is the function defined in part (a).
   2. Taking \(x _ { 1 } = 1\) as a first approximation to the root of the equation \(\mathrm { f } ( x ) = 0\), use the Newton-Raphson method to find a second approximation \(x _ { 2 }\) to the root.
      (3 marks)
3. The region enclosed by the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = 1\) and the \(x\)-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
   (3 marks)

2

1. Find \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), where \(a > 0\).
2. Write down the value of \(\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }\), where \(k\) is a positive constant.
3. Hence find \(\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).

4

1. Explain why \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) is an improper integral.
   (1 mark)
2. Use integration by parts to find \(\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\).
   (3 marks)
3. Show that \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) exists and find its value.
   (4 marks)

7

1. Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
2. Use the substitution \(u = x \mathrm { e } ^ { - x } + 1\) to find \(\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x\).
3. Hence evaluate \(\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing the limiting process used.

1 Which of the integrals below is not an improper integral?
Circle your answer. \(\int _ { 0 } ^ { \infty } e ^ { - x } d x\) \(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)

5
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.

Let $$I_n = \int_0^1 (1 + x^2)^{-n} dx,$$ where \(n \geqslant 1\). By considering \(\frac{d}{dx}(x(1 + x^2)^{-n})\), or otherwise, prove that $$2nI_{n+1} = (2n - 1)I_n + 2^{-n}.$$ [5] Deduce that \(I_3 = \frac{3}{32}\pi + \frac{1}{4}\). [3]

Use l'Hôpital's rule to show that $$\lim_{x \to \infty} (xe^{-x}) = 0$$ Fully justify your answer. [4 marks]

Evaluate the improper integral $$\int_0^8 \ln x \, \mathrm{d}x$$ showing the limiting process. [6 marks]

A student states that \(\int_0^{\pi/2} \frac{\cos x + \sin x}{\cos x - \sin x} \, dx\) is not an improper integral because \(\frac{\cos x + \sin x}{\cos x - \sin x}\) is defined at both \(x = 0\) and \(x = \frac{\pi}{2}\) Assess the validity of the student's argument. [2 marks]

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

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]
2. Show that \(\int_{-2}^{-\frac{1}{2}} f(x) \, dx = \frac{\pi\sqrt{3}}{18}\) [5 marks]
3. Find the value of \(\int_{-2}^{\infty} f(x) \, dx\) Fully justify your answer. [2 marks]

Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]

In this question you must show detailed reasoning. Evaluate \(\int_0^{\infty} 2xe^{-x} dx\). [You may use the result \(\lim_{x \to \infty} xe^{-x} = 0\).] [4]

In this question you must show detailed reasoning. Show that $$\int_5^{\infty} (x - 1)^{-\frac{3}{2}} dx = 1$$ [5]