It is given that
\(I_n=\int_0^{\frac14\pi}\frac{\sin^{2n}x}{\cos x}\,dx,\qquad n\geq0.\)
Show that
\(I_n-I_{n+1}=\frac{2^{-\left(n+\frac12\right)}}{2n+1}.\)
Hence show that
\(\int_0^{\frac14\pi}\frac{\sin^6x}{\cos x}\,dx=\ln(1+\sqrt2)-\frac{73\sqrt2}{120}.\)
It is given that \(I_{n}=\int_{1}^{\mathrm{e}}(\ln x)^{n} \mathrm{~d} x\) for \(n \geqslant 0\). Show that
\(I_{n}=(n-1)\left[I_{n-2}-I_{n-1}\right] \text { for } n \geqslant 2 \text {. }\)
Hence find, in an exact form, the mean value of \((\ln x)^{3}\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm{e}\).
Let \(I_{n}=\int_{0}^{\frac{1}{2} \pi} \frac{\sin 2 n \theta}{\cos \theta} \mathrm{~d} \theta\), where \(n\) is a non-negative integer.
(i) Use the identity \(\sin P+\sin Q \equiv 2 \sin \frac{1}{2}(P+Q) \cos \frac{1}{2}(P-Q)\) to show that \(I_{n}+I_{n-1}=\frac{2}{2 n-1}\), for all positive integers \(n\).
(ii) Find the exact value of \(\int_{0}^{\frac{1}{2} \pi} \frac{\sin 8 \theta}{\cos \theta} d \theta\).
Let \(I_{n}=\int_{0}^{\frac{1}{2} \pi} x^{n} \sin x \mathrm{~d} x\), where \(n\) is a non-negative integer. Show that
\(I_{n}=n\left(\frac{1}{2} \pi\right)^{n-1}-n(n-1) I_{n-2}, \quad \text { for } n \geqslant 2 .\)
Find the exact value of \(I_{4}\).
Evaluate \(\int_{0}^{\frac{1}{2} \pi} x \sin x \mathrm{~d} x\).
Given that \(I_{n}=\int_{0}^{\frac{1}{2} \pi} x^{n} \sin x \mathrm{~d} x\), prove that, for \(n\gt 1\),
\(I_{n}=n\left(\frac{1}{2} \pi\right)^{n-1}-n(n-1) I_{n-2}\)
By first using the substitution \(x=\cos ^{-1} u\), find the value of
\(\int_{0}^{1}\left(\cos ^{-1} u\right)^{3} \mathrm{~d} u\)
giving your answer in an exact form.
Let \(I_{n}=\int_{0}^{2} x^{n}\left(4-x^{2}\right)^{\frac{1}{2}} \mathrm{~d} x\), for \(n \geqslant 1\). By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left\{x^{n}\left(4-x^{2}\right)^{\frac{3}{2}}\right\}\), show that
\((n+3) I_{n+1}=4 n I_{n-1}, \text { where } n \geqslant 2\)
Find the value of \(I_{1}\) and deduce the exact value of \(I_{3}\).
Let \(I_{n}=\int_{0}^{\frac{1}{2} \pi} \cos ^{n} x \sin ^{2} x \mathrm{~d} x\), for \(n \geqslant 0\). By differentiating \(\cos ^{n-1} x \sin ^{3} x\) with respect to \(x\), prove that
\((n+2) I_{n}=(n-1) I_{n-2} \quad \text { for } n \geqslant 2 .\)
Hence find the exact value of \(I_{4}\).
Let \(I_{n}=\int_{0}^{\frac{1}{4} \pi} \sec ^{n} x \mathrm{~d} x\) for \(n\gt 0\).
(i) Find the value of \(I_{2}\).
(ii) Show that, for \(n\gt 2\),
\((n-1) I_{n}=2^{\frac{1}{2} n-1}+(n-2) I_{n-2} .\)
(iii) The curve \(C\) has equation \(y=\sec ^{3} x\) for \(0 \leqslant x \leqslant \frac{1}{4} \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x=\frac{1}{4} \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Let \(I_{n}=\int_{0}^{1}(1-x)^{n} \mathrm{e}^{x} \mathrm{~d} x\). Show that, for all positive integers \(n\),
\(I_{n}=n I_{n-1}-1\)
Find the exact value of \(I_{4}\).
By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y=(1-x)^{4} \mathrm{e}^{x}\) in the interval \(0 \leqslant x \leqslant 1\), show that
\(\frac{65}{24}\lt \mathrm{e}\lt \frac{11}{4} .\)
Let \(I_{n}=\int_{0}^{1}(1-x)^{n} \mathrm{e}^{x} \mathrm{~d} x\). Show that, for all positive integers \(n\),
\(I_{n}=n I_{n-1}-1 .\)
Find the exact value of \(I_{4}\).
By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y=(1-x)^{4} \mathrm{e}^{x}\) in the interval \(0 \leqslant x \leqslant 1\), show that
\(\frac{65}{24}\lt \mathrm{e}\lt \frac{11}{4} .\)
Let \(I_{n}=\int_{0}^{1} \frac{1}{\left(1+x^{2}\right)^{n}} \mathrm{~d} x\). Prove that, for every positive integer \(n\),
\(2 n I_{n+1}=2^{-n}+(2 n-1) I_{n} .\)
Given that \(I_{1}=\frac{1}{4} \pi\), find the exact value of \(I_{3}\).
Show that \(\int_{0}^{1} x \mathrm{e}^{-x^{2}} \mathrm{~d} x=\frac{1}{2}-\frac{1}{2 \mathrm{e}}\).
Let \(I_{n}=\int_{0}^{1} x^{n} \mathrm{e}^{-x^{2}} \mathrm{~d} x\). Show that \(I_{2 n+1}=n I_{2 n-1}-\frac{1}{2 \mathrm{e}}\) for \(n \geqslant 1\).
Find the exact value of \(I_{7}\).
It is given that
\(I_n=\int_0^1 \frac{x^n}{\sqrt{1+2x}}\,\mathrm{d}x\).
Show that, for \(n\geqslant 1\),
\((2n+1)I_n=\sqrt3-nI_{n-1}\).
Show that
\(I_3=\frac{2}{35}(\sqrt3+1)\).
It is given that
\(I_n=\int_0^1 \frac{x^n}{\sqrt{1+2x}}\,\mathrm{d}x\).
Show that, for \(n\geqslant 1\),
\((2n+1)I_n=\sqrt3-nI_{n-1}\).
Show that
\(I_3=\frac{2}{35}(\sqrt3+1)\).
Let \(I_{n}=\int_{0}^{1} \frac{1}{\left(1+x^{4}\right)^{n}} \mathrm{~d} x\). By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{x}{\left(1+x^{4}\right)^{n}}\right)\), show that
\(4 n I_{n+1}=\frac{1}{2^{n}}+(4 n-1) I_{n} .\)
Given that \(I_{1}=0.86697\), correct to 5 decimal places, find \(I_{3}\).
Let
\(I_{n}=\int_{1}^{\mathrm{e}} x^{2}(\ln x)^{n} \mathrm{~d} x\)
for \(n \geqslant 0\). Show that, for all \(n \geqslant 1\),
\(I_{n}=\frac{1}{3} \mathrm{e}^{3}-\frac{1}{3} n I_{n-1} .\)
Find the exact value of \(I_{3}\).
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 \(\int_{0}^{\frac{1}{4} \pi} \frac{\sin ^{6} x}{\cos x} \mathrm{~d} x=\ln (1+\sqrt{ } 2)-\frac{73}{120} \sqrt{ } 2\).
Show that \(\int x\left(1-x^{2}\right)^{\frac{1}{2}} \mathrm{~d} x=-\frac{1}{3}\left(1-x^{2}\right)^{\frac{3}{2}}+c\), where \(c\) is a constant.
Given that \(I_{n}=\int_{0}^{1} x^{n}\left(1-x^{2}\right)^{\frac{1}{2}} \mathrm{~d} x\), prove that, for \(n \geqslant 2\),
\((n+2) I_{n}=(n-1) I_{n-2} .\)
Use the substitution \(x=\sin u\) to show that
\(\int_{0}^{1}\left(1-x^{2}\right)^{\frac{1}{2}} \mathrm{~d} x=\frac{1}{4} \pi\)
Find \(I_{4}\).
Let
\(I_{n}=\int_{1}^{\mathrm{e}} x(\ln x)^{n} \mathrm{~d} x,\)
where \(n \geqslant 1\). Show that
\(I_{n+1}=\frac{1}{2} \mathrm{e}^{2}-\frac{1}{2}(n+1) I_{n} .\)
Hence prove by induction that, for all positive integers \(n, I_{n}\) is of the form \(A_{n} \mathrm{e}^{2}+B_{n}\), where \(A_{n}\) and \(B_{n}\) are rational numbers.
Let
\(I_{n}=\int_{0}^{\frac{1}{4} \pi} \tan ^{n} x \mathrm{~d} x\)
where \(n \geqslant 0\). Use the fact that \(\tan ^{2} x=\sec ^{2} x-1\) to show that, for \(n \geqslant 2\),
\(I_{n}=\frac{1}{n-1}-I_{n-2}\)
Show that \(I_{8}=\frac{1}{7}-\frac{1}{5}+\frac{1}{3}-1+\frac{1}{4} \pi\).