4.08c Improper integrals: infinite limits or discontinuous integrands

5

1. Find \(\int x ^ { 3 } \ln x \mathrm {~d} x\).
2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\) is an improper integral.
3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x\), showing the limiting process used.

4 Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \left( \frac { 1 } { x } - \frac { 4 } { 4 x + 1 } \right) \mathrm { d } x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant to be found.

3

1. Explain why \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\) is an improper integral.
2. Find \(\quad \int 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\).
3. Hence evaluate \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\), showing the limiting process used.

3

1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\) is an improper integral.
3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\), showing the limiting process used.

5

1. Find \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).
2. Hence evaluate \(\int _ { 0 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\), showing the limiting process used.

4 Evaluate the improper integral $$\int _ { 0 } ^ { \infty } \left( \frac { 2 x } { x ^ { 2 } + 4 } - \frac { 4 } { 2 x + 3 } \right) \mathrm { d } x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant.

5

1. Find \(\int x \cos 8 x \mathrm {~d} x\).
2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x } \sin 2 x \right]\).
3. Explain why \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\) is an improper integral.
4. Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\), showing the limiting process used. Give your answer as a single term.
   [0pt] [4 marks]

4

1. Explain why \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\) is an improper integral.
2. Evaluate \(\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), showing the limiting process used.

6

1. Use the substitution \(a = \frac { 1 } { p }\) to find \(\lim _ { p \rightarrow \infty } \left[ \frac { \ln p } { p ^ { k } } \right]\), where \(k > 0\).
2. Evaluate the improper integral \(\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 7 } } \mathrm {~d} x\), showing the limiting process used.
   [0pt] [4 marks]
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7

1. Determine the values of \(A , B , C\) and \(D\) such that \(\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }\).
2. In this question you must show detailed reasoning. Hence determine the exact value of \(\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x\).

6 In this question you must show detailed reasoning.
Determine the exact value of \(\int _ { 9 } ^ { \infty } \frac { 18 } { x ^ { 2 } \sqrt { x } } \mathrm {~d} x\).

2

1. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
   1. \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
   2. \(\int _ { 0 } ^ { 9 } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).
2. Explain briefly why the integrals in part (a) are improper integrals.

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

1. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 3 } { 4 } } \mathrm {~d} x\);
2. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\);
3. \(\quad \int _ { 1 } ^ { \infty } \left( x ^ { - \frac { 3 } { 4 } } - x ^ { - \frac { 5 } { 4 } } \right) \mathrm { d } x\).

2

1. Find, in terms of \(p\) and \(q\), the value of the integral \(\int _ { p } ^ { q } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).
2. Show that only one of the following improper integrals has a finite value, and find that value:
   1. \(\int _ { 0 } ^ { 2 } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\);
   2. \(\int _ { 2 } ^ { \infty } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x\).

2 Show that only one of the following improper integrals has a finite value, and find that value:

1. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 2 } { 3 } } \mathrm {~d} x\);
2. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 4 } { 3 } } \mathrm {~d} x\).

4 Show that the improper integral \(\int _ { 25 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\) has a finite value and find that value.

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

1. \(\quad \int _ { 0 } ^ { 1 } \left( x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } \right) \mathrm { d } x\);
2. \(\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x\).

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

1. \(\int _ { 9 } ^ { \infty } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\);
2. \(\int _ { 9 } ^ { \infty } \frac { 1 } { x \sqrt { x } } \mathrm {~d} x\).

5

1. A curve has equation \(y = 2 x ^ { 2 } - 5 x\).
   The point \(P\) on the curve has coordinates \(( 1 , - 3 )\).
   The point \(Q\) on the curve has \(x\)-coordinate \(1 + h\).
   1. Show that the gradient of the line \(P Q\) is \(2 h - 1\).
   2. Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point \(P\) is parallel to the line \(x + y = 0\).
2. For the improper integral \(\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x\), either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.

2

1. Explain why \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) is an improper integral.
2. Either find the value of the integral \(\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x\) or explain why it does not have a finite value.
   [0pt] [4 marks]
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6 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).

2 In this question you must show detailed reasoning. Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 2 } - 4 x + 5 } d x\)

7

1. Explain why \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt [ 3 ] { x - 2 } } \mathrm {~d} x\) is an improper integral.
2. In this question you must show detailed reasoning. Use an appropriate limit argument to evaluate this integral.

1. Show that

$$\int _ { 0 } ^ { \infty } \frac { 8 x - 12 } { \left( 2 x ^ { 2 } + 3 \right) ( x + 1 ) } \mathrm { d } x = \ln k$$ where \(k\) is a rational number to be found.

1. (a) Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x\) is an improper integral.
   (b) Prove that

$$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } d x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.