1.08i Integration by parts

$$I_n = \int (x^2 + 1)^{-n} dx, \quad n > 0$$

1. Show that, for \(n > 0\) $$I_{n+1} = \frac{x(x^2 + 1)^{-n}}{2n} + \frac{2n - 1}{2n}I_n$$ [5]
2. Find \(I_2\) [3]

$$I_n = \int_0^{\frac{\pi}{2}} x^n \cos x \, dx, \quad n \geq 0.$$

1. Prove that \(I_n = \left(\frac{\pi}{2}\right)^n - n(n-1)I_{n-2}\), \(n \geq 2\). [5]
2. Find an exact expression for \(I_6\). [4]

$$I_n = \int_0^1 x^n e^x \, dx \text{ and } J_n = \int_0^1 x^n e^{-x} \, dx, \quad n \geq 0.$$

1. Show that, for \(n \geq 1\), $$I_n = e - nI_{n-1}.$$ [2]
2. Find a similar reduction formula for \(J_n\). [3]
3. Show that \(J_2 = 2 - \frac{5}{e}\). [3]
4. Show that \(\int_0^1 x^n \cosh x \, dx = \frac{1}{2}(I_n + J_n)\). [1]
5. Hence, or otherwise, evaluate \(\int_0^1 x^2 \cosh x \, dx\), giving your answer in terms of \(e\). [4]

$$I_n = \int_0^1 x^n e^{2x} \, dx, \quad n \geq 0.$$

1. Prove that, for \(n \geq 1\), $$I_n = \frac{1}{2}(x^n e^{2x} - nI_{n-1}).$$ [3]
2. Find, in terms of \(e\), the exact value of $$\int_0^1 x^2 e^{2x} \, dx.$$ [5]

\includegraphics{figure_33} Figure 2 shows a sketch of the curve with equation $$y = x \operatorname{arcosh} x, \quad 1 \leq x \leq 2.$$ The region \(R\), as shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = 2\). Show that the area of \(R\) is $$\frac{7}{4} \ln(2 + \sqrt{3}) - \frac{\sqrt{3}}{2}.$$ (Total 10 marks)

$$I_n = \int_0^{\pi} x^n \sin x \, dx$$

1. Show that for \(n \geq 2\) $$I_n = n \left( \frac{\pi}{2} \right)^{n-1} - n(n-1)I_{n-2}$$ [4]
2. Hence obtain \(I_3\), giving your answers in terms of \(\pi\). [4]

(Total 8 marks)

Fig. 8 shows part of the curve \(y = x \sin 3x\). It crosses the \(x\)-axis at P. The point on the curve with \(x\)-coordinate \(\frac{1}{6}\pi\) is Q. \includegraphics{figure_8}

1. Find the \(x\)-coordinate of P. [3]
2. Show that Q lies on the line \(y = x\). [1]
3. Differentiate \(x \sin 3x\). Hence prove that the line \(y = x\) touches the curve at Q. [6]
4. Show that the area of the region bounded by the curve and the line \(y = x\) is \(\frac{1}{72}(\pi^2 - 8)\). [7]

Show that \(\int_0^{\frac{\pi}{2}} x \cos \frac{1}{2} x \, dx = \frac{\sqrt{2}}{2} \pi + 2\sqrt{2} - 4\). [5]

Find \(\int_{-1}^4 x^{-\frac{1}{2}} \ln x dx\), giving your answer in an exact form. [5]

Find \(\int x \sin 3x dx\). [4]

Fig. 8 shows the line \(y = x\) and parts of the curves \(y = f(x)\) and \(y = g(x)\), where $$f(x) = e^{x-1}, \quad g(x) = 1 + \ln x.$$ 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. \includegraphics{figure_8}

1. Find the exact coordinates of A and B. Verify that the coordinates of C are \((1, 1)\). [5]
2. Prove algebraically that \(g(x)\) is the inverse of \(f(x)\). [2]
3. Evaluate \(\int_0^1 f(x) \, dx\), giving your answer in terms of \(e\). [3]
4. Use integration by parts to find \(\int \ln x \, dx\). Hence show that \(\int_{e^{-1}}^1 g(x) \, dx = \frac{1}{e}\). [6]
5. Find the area of the region enclosed by the lines OA and OB, and the arcs AC and BC. [2]

A curve is defined by the equation \(y = 2x \ln(1 + x)\).

1. Find \(\frac{dy}{dx}\) and hence verify that the origin is a stationary point of the curve. [4]
2. Find \(\frac{d^2y}{dx^2}\) and use this to verify that the origin is a minimum point. [5]
3. Using the substitution \(u = 1 + x\), show that \(\int \frac{x^2}{1+x} \, dx = \int \left(u - 2 + \frac{1}{u}\right) du\). Hence evaluate \(\int_0^1 \frac{x^2}{1+x} \, dx\), giving your answer in an exact form. [6]
4. Using integration by parts and your answer to part (iii), evaluate \(\int_0^1 2x \ln(1 + x) \, dx\). [4]

Find \(\int xe^{3x} \, dx\). [4]

Use integration by parts to find the exact value of \(\int_1^3 x^2 \ln x \, dx\). [6]

1. Use integration by parts to show that $$\int_0^{\frac{\pi}{4}} x \sec^2 x \, dx = \frac{1}{4}\pi - \frac{1}{2} \ln 2.$$ [6]

\includegraphics{figure_1} The finite region \(R\), bounded by the equation \(y = x^{\frac{1}{2}} \sec x\), the line \(x = \frac{\pi}{4}\) and the \(x\)-axis is shown in Fig. 1. The region \(R\) is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Find the volume of the solid of revolution generated. [2]
2. Find the gradient of the curve with equation \(y = x^{\frac{1}{2}} \sec x\) at the point where \(x = \frac{\pi}{4}\). [3]

Find the exact value of \(\int_1^2 x \ln x \, dx\). [5]

Evaluate \(\int_0^{\frac{\pi}{2}} x \cos x dx\), giving your answer in an exact form. [5]

Find \(\int xe^{3x} dx\). [4]