1.08i Integration by parts

10 \includegraphics[max width=\textwidth, alt={}, center]{0f081749-4fe0-46e3-96c2-466e69cf49d3-4_620_894_338_687} The function f is defined by \(\mathrm { f } ( x ) = ( \ln x ) ^ { 2 }\) for \(x > 0\). The diagram shows a sketch of the graph of \(y = \mathrm { f } ( x )\). The minimum point of the graph is \(A\). The point \(B\) has \(x\)-coordinate e .

1. State the \(x\)-coordinate of \(A\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(B\).
3. Use the substitution \(x = \mathrm { e } ^ { u }\) to show that the area of the region bounded by the \(x\)-axis, the line \(x = \mathrm { e }\), and the part of the curve between \(A\) and \(B\) is given by $$\int _ { 0 } ^ { 1 } u ^ { 2 } \mathrm { e } ^ { u } \mathrm {~d} u .$$
4. Hence, or otherwise, find the exact value of this area.

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

10 \includegraphics[max width=\textwidth, alt={}, center]{2718ebbb-29e3-46f7-8d8d-ec7d526483f8-3_458_920_1144_609} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 2 } }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\).

1. Write down the \(x\)-coordinate of \(A\).
2. Find the exact coordinates of \(M\).
3. Use integration by parts to find the exact area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = \mathrm { e }\).

2 Show that \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x = \pi ^ { 2 } - 4\).

7 The integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } 4 t ^ { 3 } \ln \left( t ^ { 2 } + 1 \right) \mathrm { d } t\).

1. Use the substitution \(x = t ^ { 2 } + 1\) to show that \(I = \int _ { 1 } ^ { 5 } ( 2 x - 2 ) \ln x \mathrm {~d} x\).
2. Hence find the exact value of \(I\).

10 \includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-3_451_933_1777_605} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

1. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\) is equal to \(2 - \frac { 17 } { \mathrm { e } ^ { 3 } }\).
2. Find the \(x\)-coordinate of the maximum point \(M\) on the curve.
3. Find the \(x\)-coordinate of the point \(P\) at which the tangent to the curve passes through the origin.

3 Show that \(\int _ { 0 } ^ { 1 } ( 1 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 4 \mathrm { e } ^ { - \frac { 1 } { 2 } } - 2\).

9 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689} The diagram shows the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\). The shaded region between the curve, the \(x\)-axis and the line \(x = \mathrm { e }\) is denoted by \(R\).

1. Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = m x + c\).
2. Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.

8

1. Show that \(\int _ { 2 } ^ { 4 } 4 x \ln x \mathrm {~d} x = 56 \ln 2 - 12\).
2. Use the substitution \(u = \sin 4 x\) to find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 24 } \pi } \cos ^ { 3 } 4 x \mathrm {~d} x\).

9 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { 2 - x }\) and its maximum point \(M\).

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

6 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).

1. Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
2. Verify by calculation that \(a\) is greater than 1 .
3. Use the iterative formula $$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.

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

3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { 2 } \sin 2 x \mathrm {~d} x\).

4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta\).

2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x ^ { 2 } \cos 2 x \mathrm {~d} x = \frac { 1 } { 32 } \left( \pi ^ { 2 } - 8 \right)\).

10 \includegraphics[max width=\textwidth, alt={}, center]{e26f21c5-3776-4c86-8440-6959c5e37486-18_337_529_260_808} The diagram shows the curve \(y = ( \ln x ) ^ { 2 }\). The \(x\)-coordinate of the point \(P\) is equal to e, and the normal to the curve at \(P\) meets the \(x\)-axis at \(Q\).

1. Find the \(x\)-coordinate of \(Q\).
2. Show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is a constant.
3. Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the \(x\)-axis and the normal \(P Q\).

4 Show that \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 3 } { 2 } } \ln x \mathrm {~d} x = 2 - \ln 4\).

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

7 \includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-3_480_901_973_621} The diagram shows the curve \(y = x ^ { 2 } e ^ { - \frac { 1 } { 2 } x }\).

1. Find the \(x\)-coordinate of \(M\), the maximum point of the curve.
2. Find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 1\), giving your answer in terms of e.

9 The complex number \(u\) is given by $$u = \frac { 3 + \mathrm { i } } { 2 - \mathrm { i } }$$

1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Find the modulus and argument of \(u\).
3. Sketch an Argand diagram showing the point representing the complex number \(u\). Show on the same diagram the locus of the point representing the complex number \(z\) such that \(| z - u | = 1\).
4. Using your diagram, calculate the least value of \(| z |\) for points on this locus.

2 The polynomial \(x ^ { 4 } + 3 x ^ { 2 } + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } + x + 2\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\) and the other quadratic factor of \(\mathrm { p } ( x )\).

9 The constant \(a\) is such that \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x = 6\).

1. Show that \(a\) satisfies the equation $$x = 2 + \mathrm { e } ^ { - \frac { 1 } { 2 } x } .$$
2. By sketching a suitable pair of graphs, show that this equation has only one root.
3. Verify by calculation that this root lies between 2 and 2.5.
4. Use an iterative formula based on the equation in part (i) to calculate the value of \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9 \includegraphics[max width=\textwidth, alt={}, center]{8d134c65-af23-4508-acef-49b6ab49e374-3_504_910_625_614} The diagram shows the curve \(y = \frac { \ln x } { \sqrt { } x }\) and its maximum point \(M\). The curve cuts the \(x\)-axis at the point \(A\).

1. State the coordinates of \(A\).
2. Find the exact value of the \(x\)-coordinate of \(M\).
3. Using integration by parts, show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 4\) is equal to \(8 \ln 2 - 4\).

9 \includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).

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

9 \includegraphics[max width=\textwidth, alt={}, center]{822f851a-7fae-43b8-9ebc-94588f01e51c-4_597_895_258_625} The diagram shows the curve \(y = x ^ { 3 } \ln x\) and its minimum point \(M\).

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