1.08i Integration by parts

7

1. Given that \(\int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = \frac { 2 } { 5 }\), show that \(a = \frac { 5 } { 3 } ( 1 + \ln a )\).
2. Use an iteration formula based on the equation \(a = \frac { 5 } { 3 } ( 1 + \ln a )\) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{f421f03c-57c9-4feb-91b9-a7f9b12f96ce-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).

1. Find the exact values of the coordinates of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{9e129863-5994-4e17-81f8-e139515998d1-3_666_956_1231_593} The diagram shows the curve \(y = x ^ { 2 } \ln x\) and its minimum point \(M\).

1. Find the exact values of the coordinates of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).

5 It is given that \(\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22\), where \(a\) is a constant greater than 1 .

1. Show that \(a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)\).
2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.

5 The expression \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x \mathrm { e } ^ { - 2 x }\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( - \frac { 1 } { 2 } \right)\).
2. Find the exact value of \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x\).

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

5 It is given that \(\int _ { 0 } ^ { p } 4 x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 9\), where \(p\) is a positive constant.

1. Show that \(p = 2 \ln \left( \frac { 8 p + 16 } { 7 } \right)\).
2. Use an iterative process based on the equation in part (i) to find the value of \(p\) correct to 3 significant figures. Use a starting value of 3.5 and give the result of each iteration to 5 significant figures.

6 It is given that \(\int _ { 1 } ^ { a } \ln ( 2 x ) \mathrm { d } x = 1\), where \(a > 1\).

1. Show that \(a = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a } \right)\), where \(\exp ( x )\) denotes \(\mathrm { e } ^ { x }\).
2. Use the iterative formula $$a _ { n + 1 } = \frac { 1 } { 2 } \exp \left( 1 + \frac { \ln 2 } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7 \includegraphics[max width=\textwidth, alt={}, center]{f4614578-f5f6-4283-8185-8b5598ad91d5-3_416_679_258_731} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).

1. Find the exact \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.

7 \includegraphics[max width=\textwidth, alt={}, center]{84df6b9a-6118-44a2-9c18-512039ded4fd-3_416_677_258_733} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).

1. Find the exact \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.

9 It is given that \(\int _ { 1 } ^ { a } x ^ { \frac { 1 } { 2 } } \ln x \mathrm {~d} x = 2\), where \(a > 1\).

1. Show that \(a ^ { \frac { 3 } { 2 } } = \frac { 7 + 2 a ^ { \frac { 3 } { 2 } } } { 3 \ln a }\).
2. Show by calculation that \(a\) lies between 2 and 4 .
3. Use the iterative formula $$a _ { n + 1 } = \left( \frac { 7 + 2 a _ { n } ^ { \frac { 3 } { 2 } } } { 3 \ln a _ { n } } \right) ^ { \frac { 2 } { 3 } }$$ to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-16_446_956_260_593} The diagram shows the curve \(y = \left( 1 + x ^ { 2 } \right) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).

1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve.
2. Show that the exact value of the area of \(R\) is \(18 - \frac { 42 } { \mathrm { e } }\).

6

1. By differentiating \(\frac { \cos x } { \sin x }\), show that if \(y = \cot x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
2. Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 4 } ( \pi + \ln 4 )\). \(7 \quad\) Two lines \(l\) and \(m\) have equations \(\mathbf { r } = a \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) respectively, where \(a\) is a constant. It is given that the lines intersect.

9 It is given that \(\int _ { 0 } ^ { a } x \cos \frac { 1 } { 3 } x \mathrm {~d} x = 3\), where the constant \(a\) is such that \(0 < a < \frac { 3 } { 2 } \pi\).

1. Show that \(a\) satisfies the equation $$a = \frac { 4 - 3 \cos \frac { 1 } { 3 } a } { \sin \frac { 1 } { 3 } a }$$
2. Verify by calculation that \(a\) lies between 2.5 and 3 .
3. Use an iterative formula based on the equation in part (i) to calculate \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

8 \includegraphics[max width=\textwidth, alt={}, center]{dde12c57-5129-43ae-b385-9a8f21f51e49-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

8 \includegraphics[max width=\textwidth, alt={}, center]{2aceb797-097c-499b-99b6-cce9f287cb51-3_566_787_255_680} The diagram shows the curve \(y = x \sin x\), for \(0 \leqslant x \leqslant \pi\). The point \(Q \left( \frac { 1 } { 2 } \pi , \frac { 1 } { 2 } \pi \right)\) lies on the curve.

1. Show that the normal to the curve at \(Q\) passes through the point \(( \pi , 0 )\).
2. Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin x - x \cos x )\).
3. Hence evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).

3 Find the exact value of $$\int _ { 1 } ^ { 4 } x ^ { \frac { 3 } { 2 } } \ln x \mathrm {~d} x$$

2 Find the exact value of \(\int _ { 0 } ^ { 1 } ( 2 - x ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).

9 The equation of a curve is \(y = x ^ { - \frac { 2 } { 3 } } \ln x\) for \(x > 0\). The curve has one stationary point.

1. Find the exact coordinates of the stationary point.
2. Show that \(\int _ { 1 } ^ { 8 } y \mathrm {~d} x = 18 \ln 2 - 9\).

4 Using integration by parts, find the exact value of \(\int _ { 0 } ^ { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).

8 \includegraphics[max width=\textwidth, alt={}, center]{1990cbac-d96f-4484-be4b-67dab35b3147-12_458_725_262_708} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 4 } }\) and its maximum point \(M\).

1. Find the exact coordinates of \(M\).
2. By using integration by parts, show that for all \(a > 1 , \int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 4 } } \mathrm {~d} x < \frac { 1 } { 9 }\).

10 The constant \(a\) is such that \(\int _ { 1 } ^ { a } x ^ { 2 } \ln x \mathrm {~d} x = 4\).

1. Show that \(a = \left( \frac { 35 } { 3 \ln a - 1 } \right) ^ { \frac { 1 } { 3 } }\).
2. Verify by calculation that \(a\) lies between 2.4 and 2.8.
3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x = \frac { 1 } { 8 }\).

1. Show that \(a = \frac { 1 } { 2 } \ln ( 4 a + 2 )\).
2. Verify by calculation that \(a\) lies between 0.5 and 1 .
3. Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

7

1. Use the substitution \(u = \cos x\) to show that $$\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x = \int _ { - 1 } ^ { 1 } 2 u \mathrm { e } ^ { 2 u } \mathrm {~d} u$$
2. Hence find the exact value of \(\int _ { 0 } ^ { \pi } \sin 2 x \mathrm { e } ^ { 2 \cos x } \mathrm {~d} x\).

10

1. Given that \(2 x = \tan y\), show that \(\frac { d y } { d x } = \frac { 2 } { 1 + 4 x ^ { 2 } }\).
2. Hence find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { \frac { \sqrt { 3 } } { 2 } } x \tan ^ { - 1 } ( 2 x ) \mathrm { d } x\).