4.08c Improper integrals: infinite limits or discontinuous integrands

10 The function f is defined by \(\mathrm { f } ( x ) = ( 4 x + 2 ) ^ { - 2 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
   A point is moving along the curve \(y = \mathrm { f } ( x )\) in such a way that, as it passes through the point \(A\), its \(y\)-coordinate is decreasing at the rate of \(k\) units per second and its \(x\)-coordinate is increasing at the rate of \(k\) units per second.
2. Find the coordinates of \(A\).

1 Find the exact value of \(\int _ { 3 } ^ { \infty } \frac { 2 } { x ^ { 2 } } d x\).

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { ( x + 2 ) ^ { 2 } }\) for \(x > - 2\).

1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
2. The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x )\). It is given that the point \(( - 1 , - 1 )\) lies on the curve. Find the equation of the curve.

10

1. Find \(\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\). \includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726} The diagram shows the curve with equation \(y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). The shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the volume of revolution.
   The normal to the curve at the point \(( 1,1 )\) crosses the \(y\)-axis at the point \(A\).
3. Find the \(y\)-coordinate of \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2

1. Find \(\int _ { 0 } ^ { a } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\), where \(a\) is a positive constant.
2. Deduce the value of \(\int _ { 0 } ^ { \infty } \left( \mathrm { e } ^ { - x } + 6 \mathrm { e } ^ { - 3 x } \right) \mathrm { d } x\).

4 Using the substitution \(u = \sqrt { x }\), find the exact value of $$\int _ { 3 } ^ { \infty } \frac { 1 } { ( x + 1 ) \sqrt { x } } \mathrm {~d} x$$

6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\).
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-11_78_1576_336_321}

6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { 1 }\).
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined. \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-11_78_1576_336_321}

7 The integral \(\mathrm { I } _ { \mathrm { n } }\), where n is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 4 } { 3 } } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).

1. Find the exact value of \(I _ { - 1 }\) giving your answer in the form \(\ln a\), where \(a\) is an integer to be determined.
2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm { n } \right)\), or otherwise, show that $$( \mathrm { n } + 1 ) \mathrm { I } _ { \mathrm { n } } = \mathrm { nl } _ { \mathrm { n } - 2 } + \frac { 4 } { 3 } \left( \frac { 5 } { 3 } \right) ^ { \mathrm { n } }$$
3. A curve has equation \(y = x ^ { 2 }\), for \(0 \leqslant x \leqslant \frac { 2 } { 3 }\). The arc length of the curve is denoted by \(s\). Use the substitution \(\mathrm { u } = 2 \mathrm { x }\) to show that \(\mathrm { s } = \frac { 1 } { 2 } \mathrm { l } _ { 1 }\) and find the exact value of \(s\).

6 \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-12_533_1532_278_264} The diagram shows the curve with equation \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\).

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x > L _ { N }\), where $$L _ { N } = \frac { 1 } { 2 N \left( 2 ^ { \frac { 1 } { N } } - 1 \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-12_2717_38_109_2009}
2. Use a similar method to find, in terms of \(N\), an upper bound \(U _ { N }\) for \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(U _ { N } - L _ { N } \leqslant 10 ^ { - 3 }\).
4. Given that \(\int _ { 0 } ^ { 1 } \left( \frac { 1 } { 2 } \right) ^ { x } \mathrm {~d} x = \frac { 1 } { 2 \ln 2 }\) ,use the value of \(N\) found in part(c)to find upper and lower bounds for \(\ln 2\) .

7. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 3 a ^ { n - 1 } } { n b ^ { n } } + \frac { n - 1 } { n } I _ { n - 2 }$$ where \(a\) and \(b\) are integers to be found.
2. Hence, or otherwise, find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 4 } x \mathrm {~d} x$$

6. Evaluate

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

4

1. Find \(\int \left( 5 x ^ { 3 } - 6 x + 1 \right) \mathrm { d } x\).
2. 1. Find \(\int 24 x ^ { - 3 } \mathrm {~d} x\).
   2. Given that \(\int _ { a } ^ { \infty } 24 x ^ { - 3 } \mathrm {~d} x = 3\), find the value of the positive constant \(a\).

5

1. Find \(\int \left( x ^ { 2 } + 2 \right) ( 2 x - 3 ) \mathrm { d } x\).
2. 1. Find, in terms of \(a\), the value of \(\int _ { 1 } ^ { a } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\), where \(a\) is a constant greater than 1 .
   2. Deduce the value of \(\int _ { 1 } ^ { \infty } \left( 6 x ^ { - 2 } - 4 x ^ { - 3 } \right) \mathrm { d } x\).

7 Expand \(\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }\) and, by substituting \(z = \cos \theta + \mathrm { i } \sin \theta\), find integers \(p , q , r , s\) such that $$64 \sin ^ { 2 } \theta \cos ^ { 4 } \theta = p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta$$ Using the substitution \(x = 2 \cos \theta\), show that $$\int _ { 1 } ^ { 2 } x ^ { 4 } \sqrt { } \left( 4 - x ^ { 2 } \right) \mathrm { d } x = \frac { 4 } { 3 } \pi + \sqrt { } 3$$

The curve \(C\) has equation \(y = 2 \sec x\), for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). Show that the arc length \(s\) of \(C\) is given by $$S = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 2 } x - 1 \right) d x$$ Find the exact value of \(s\). The surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\). Show that

1. \(S = 4 \pi \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 3 } x - \sec x \right) \mathrm { d } x\),
2. \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x \tan x ) = 2 \sec ^ { 3 } x - \sec x\). Hence find the exact value of \(S\).

10 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 2 n } x } { \cos x } \mathrm {~d} x\), where \(n \geqslant 0\). Show that $$I _ { n } - I _ { n + 1 } = \frac { 2 ^ { - \left( n + \frac { 1 } { 2 } \right) } } { 2 n + 1 }$$ Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 6 } x } { \cos x } \mathrm {~d} x = \ln ( 1 + \sqrt { } 2 ) - \frac { 73 } { 120 } \sqrt { } 2\).

5 Let \(I _ { n }\) denote \(\int _ { 0 } ^ { \infty } x ^ { n } \mathrm { e } ^ { - 2 x } \mathrm {~d} x\). Show that \(I _ { n } = \frac { 1 } { 2 } n I _ { n - 1 }\), for \(n \geqslant 1\). Prove by mathematical induction that, for all positive integers \(n , I _ { n } = \frac { n ! } { 2 ^ { n + 1 } }\).

4

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

4

1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is an arbitrary constant.
2. Hence evaluate \(\int _ { 0 } ^ { 1 } \ln x \mathrm {~d} x\), showing the limiting process used.

6

1. Explain why \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) is an improper integral.
2. 1. Show that the substitution \(y = \frac { 1 } { x }\) transforms \(\int \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) into \(\int 2 y \ln y \mathrm {~d} y\).
   2. Evaluate \(\int _ { 0 } ^ { 1 } 2 y \ln y \mathrm {~d} y\), showing the limiting process used.
   3. Hence write down the value of \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\).

5

1. Write \(\frac { 4 } { 4 x + 1 } - \frac { 3 } { 3 x + 2 }\) in the form \(\frac { C } { ( 4 x + 1 ) ( 3 x + 2 ) }\), where \(C\) is a constant.
   (l mark)
2. Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { 10 } { ( 4 x + 1 ) ( 3 x + 2 ) } d x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant.
   (6 marks)

5

1. Explain why \(\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x\) is an improper integral.
   (1 mark)
2. By using the substitution \(u = x ^ { 2 } \mathrm { e } ^ { - 4 x } + 3\), find $$\int \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$$
3. Hence evaluate \(\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x\), showing the limiting process used.

4

1. Explain why \(\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x\) is an improper integral.
   (l mark)
2. Evaluate \(\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x\), showing the limiting process used.
   (6 marks)

5

1. Show that \(\lim _ { a \rightarrow \infty } \left( \frac { 3 a + 2 } { 2 a + 3 } \right) = \frac { 3 } { 2 }\).
2. Evaluate \(\int _ { 1 } ^ { \infty } \left( \frac { 3 } { 3 x + 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\), giving your answer in the form \(\ln k\), where \(k\) is a rational number.
   (5 marks)