1.08i Integration by parts

1. (a) Use the substitution \(x = \sec \theta\) to show that

$$\int \sqrt { } \left( x ^ { 2 } - 1 \right) d x$$ can be written as $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta$$ (3)
(b) Use integration by parts to show that $$\int \sec \theta \tan ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \sec \theta \tan \theta - \ln | \sec \theta + \tan \theta | ] + \text { constant. }$$ (c) Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin x \sqrt { } ( \cos 2 x ) \mathrm { d } x\).

5.In this question u and v are functions of \(x\) .Given that \(\int \mathrm { u } \mathrm { d } x , \int \mathrm { v } \mathrm { d } x\) and \(\int \mathrm { uv } \mathrm { d } x\) satisfy $$\int \text { uv } \mathrm { d } x = \left( \int \mathrm { u } \mathrm {~d} x \right) \times \left( \int \mathrm { v } \mathrm {~d} x \right) \quad \text { uv } \neq 0$$

1. show that \(1 = \frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } + \frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) Given also that \(\frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } = \mathrm { sin } ^ { 2 } x\),
2. use part(a)to write down an expression,in terms of \(x\) ,for \(\frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) ,
3. show that $$\frac { 1 } { \mathrm { u } } \frac { \mathrm { du } } { \mathrm {~d} x } = \frac { 1 - 2 \sin x \cos x } { \sin ^ { 2 } x }$$
4. hence use integration to show that \(\mathrm { u } = A \mathrm { e } ^ { - \cot x } \operatorname { cosec } ^ { 2 } x\) ,where \(A\) is an arbitrary constant.
5. By differentiating \(\mathrm { e } ^ { \tan x }\) find a similar expression for v .

6.(a)Show that $$\frac { \mathrm { d } } { \mathrm {~d} u } \ln \left( u + \sqrt { u ^ { 2 } - 1 } \right) = \frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$$ (b)Use the result from part(a)and the substitution \(x + 3 = \frac { 1 } { t }\) to find $$\int \frac { 1 } { ( x + 3 ) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ (6)
(c)Express \(\frac { 1 } { 2 x ^ { 2 } + 13 x + 21 }\) in partial fractions.
(d)Find $$\int _ { 1 } ^ { 9 } \frac { 1 } { \left( 2 x ^ { 2 } + 13 x + 21 \right) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ giving your answer in the form \(\ln r - s\) where \(r\) and \(s\) are rational numbers.

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with equation \(y = x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - 34 x\) and the line \(L\) with equation \(y = m x + c\) . The line \(L\) touches \(C\) at the points \(P\) and \(Q\) with \(x\) coordinates \(p\) and \(q\) respectively.

1. Explain why $$x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - ( 34 + m ) x - c = ( x - p ) ^ { 2 } ( x - q ) ^ { 2 }$$ The finite region \(R\) ,shown shaded in Figure 3,is bounded by \(C\) and \(L\) .
2. Use integration by parts to show that the area of \(R\) is \(\frac { ( q - p ) ^ { 5 } } { 30 }\)
3. Show that $$( x - p ) ^ { 2 } ( x - q ) ^ { 2 } = x ^ { 4 } - 2 ( p + q ) x ^ { 3 } + S x ^ { 2 } - T x + U$$ where \(S , T\) and \(U\) are expressions to be found in terms of \(p\) and \(q\) .
4. Using part(a)and part(c)find the value of \(p\) ,the value of \(q\) and the equation of \(L\) .

5.Use integration by parts to find the exact value of $$\int _ { 0 } ^ { 2 } x 2 ^ { x } \mathrm {~d} x$$ Write your answer as a single simplified fraction.

4

1. Define tanh \(t\) in terms of exponential functions. Sketch the graph of \(\tanh t\).
2. Show that \(\operatorname { artanh } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). State the set of values of \(x\) for which this equation is valid.
3. Differentiate the equation \(\tanh y = x\) with respect to \(x\) and hence show that the derivative of \(\operatorname { artanh } x\) is \(\frac { 1 } { 1 - x ^ { 2 } }\). Show that this result may also be obtained by differentiating the equation in part (ii).
4. By considering \(\operatorname { artanh } x\) as \(1 \times \operatorname { artanh } x\) and using integration by parts, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \operatorname { artanh } x \mathrm {~d} x = \frac { 1 } { 4 } \ln \frac { 27 } { 16 }$$

1

1. 1. Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
   2. By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
   3. Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
2. 1. A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
   2. Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.

4

1. Starting with the relationship \(\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1\), deduce a relationship between \(\tanh ^ { 2 } t\) and \(\operatorname { sech } ^ { 2 } t\). You are given that \(y = \operatorname { artanh } x\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 - x ^ { 2 } }\).
3. Show, by integrating the result in part (ii), that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
4. Show that \(\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 6 } } \frac { 1 } { 1 - 3 x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { \sqrt { 3 } } \operatorname { artanh } \frac { 1 } { 2 }\). Express this answer in logarithmic form.
5. Use integration by parts to find \(\int \operatorname { artanh } x \mathrm {~d} x\), giving your answer in terms of logarithms. \section*{END OF QUESTION PAPER}

5

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
2. Show that \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).

2

1. Differentiate \(x \cos 2 x\) with respect to \(x\).
2. Integrate \(x \cos 2 x\) with respect to \(x\).

8 Fig. 8 shows the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\). P is the point on this curve with \(x\)-coordinate 1 , and R is the point \(\left( 0 , - \frac { 7 } { 8 } \right)\). \begin{figure}[h] \end{figure}

1. Find the gradient of PR.
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that PR is a tangent to the curve.
3. Find the exact coordinates of the turning point Q .
4. Differentiate \(x \ln x - x\). Hence, or otherwise, show that the area of the region enclosed by the curve \(y = x ^ { 2 } - \frac { 1 } { 8 } \ln x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is \(\frac { 59 } { 24 } - \frac { 1 } { 4 } \ln 2\).

8 Fig. 8 shows part of the curve \(y = x \cos 3 x\).
The curve crosses the \(x\)-axis at \(\mathrm { O } , \mathrm { P }\) and Q . \begin{figure}[h] \end{figure}

1. Find the exact coordinates of P and Q .
2. Find the exact gradient of the curve at the point P . Show also that the turning points of the curve occur when \(x \tan 3 x = \frac { 1 } { 3 }\).
3. Find the area of the region enclosed by the curve and the \(x\)-axis between O and P , giving your answer in exact form.

8 Fig. 8 shows the line \(y = x\) and parts of the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), where $$\mathrm { f } ( x ) = \mathrm { e } ^ { x - 1 } , \quad \mathrm {~g} ( x ) = 1 + \ln x$$ The curves intersect the axes at the points A and B , as shown. The curves and the line \(y = x\) meet at the point C . \begin{figure}[h] \end{figure}

1. Find the exact coordinates of A and B . Verify that the coordinates of C are \(( 1,1 )\).
2. Prove algebraically that \(\mathrm { g } ( x )\) is the inverse of \(\mathrm { f } ( x )\).
3. Evaluate \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in terms of e.
4. Use integration by parts to find \(\int \ln x \mathrm {~d} x\). Hence show that \(\int _ { \mathrm { e } ^ { - 1 } } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x = \frac { 1 } { \mathrm { e } }\).
5. Find the area of the region enclosed by the lines OA and OB , and the arcs AC and BC .

8 Fig. 8 shows a sketch of part of the curve \(y = x \sin 2 x\), where \(x\) is in radians.
The curve crosses the \(x\)-axis at the point P . The tangent to the curve at P crosses the \(y\)-axis at Q . \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that the \(x\)-coordinates of the turning points of the curve satisfy the equation \(\tan 2 x + 2 x = 0\).
2. Find, in terms of \(\pi\), the \(x\)-coordinate of the point P . Show that the tangent PQ has equation \(2 \pi x + 2 y = \pi ^ { 2 }\).
   Find the exact coordinates of Q .
3. Show that the exact value of the area shaded in Fig. 8 is \(\frac { 1 } { 8 } \pi \left( \pi ^ { 2 } - 2 \right)\).

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

2 Find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).

8

1. State the derivative of \(\mathrm { e } ^ { \cos x }\).
2. Hence use integration by parts to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos x \sin x \mathrm { e } ^ { \cos x } \mathrm {~d} x$$

7 Show that \(\int _ { 0 } ^ { \pi } \left( x ^ { 2 } + 5 x + 7 \right) \sin x \mathrm {~d} x = \pi ^ { 2 } + 5 \pi + 10\).

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

1 Find \(\int x \cos 3 x \mathrm {~d} x\).

4

1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).

9

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x - x ) = \ln x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8492b214-aaac-4354-8649-e317bf7b3535-3_481_725_1064_751} In the diagram, \(C\) is the curve \(y = \ln x\). The region \(R\) is bounded by \(C\), the \(x\)-axis and the line \(x = \mathrm { e }\).
   1. Find the exact volume of the solid of revolution formed by rotating \(R\) completely about the \(x\)-axis.
   2. The region \(R\) is rotated completely about the \(y\)-axis. Explain why the volume of the solid of revolution formed is given by $$\pi \mathrm { e } ^ { 2 } - \pi \int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 y } \mathrm {~d} y ,$$ and find this volume.

2 Use integration by parts to find \(\int \ln ( x + 2 ) \mathrm { d } x\).

2 Find \(\int x ^ { 8 } \ln ( 3 x ) \mathrm { d } x\).

8

1. Use division to show that \(\frac { t ^ { 3 } } { t + 2 } \equiv t ^ { 2 } - 2 t + 4 - \frac { 8 } { t + 2 }\).
2. Find \(\int _ { 1 } ^ { 2 } 6 t ^ { 2 } \ln ( t + 2 ) \mathrm { d } t\). Give your answer in the form \(A + B \ln 3 + C \ln 4\).