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Chapter 2: Problem 4
If \(f^{\prime}\) is continuous on \([a, b]\), show that \(2 \int_{a}^{b} f(x) f^{\prime}(x) d x=[f(b)]^{2}-[f(a)]^{2}\)
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Evaluate the following integrals : (i) \(\int_{0}^{1}\left(1-x^{2}\right)^{n} d x\) (ii) \(\int_{0}^{1} \frac{x^{2 n} d x}{\sqrt{1-x^{2}}}\) (iii) \(\int_{0}^{2 \mathrm{a}} \mathrm{x}^{9 / 2}(2 \mathrm{a}-\mathrm{x})^{-1 / 2} \mathrm{dx}\) (iv) \(\int_{0}^{\infty} \frac{x^{4} d x}{\left(a^{2}+x^{2}\right)^{2}}\)
Show that \(\int_{0}^{\infty} \mathrm{e}^{-x^{2}} \mathrm{dx}=\int_{0}^{1} \sqrt{-\ell \text { ny }}\) dy by interpreting th -
Evaluate the following integrals: (i) \(\int_{-\infty}^{\infty} \frac{x d x}{x^{4}+1}\) (ii) \(\int_{0}^{1} \frac{\ln (1-x)}{x} \mathrm{dx}\) (iii) \(\int_{0}^{\infty} \frac{\mathrm{dx}}{(\mathrm{x}+1)(\mathrm{x}+2)}\) (iv) \(\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}, a, b>0 .\)
Show that \(\int_{0}^{1} \frac{\ell n\left(1-a^{2} x^{2}\right)}{x^{2} \sqrt{\left(1-x^{2}\right)}} d x\) \(=\pi\left[\sqrt{1-a^{2}}-1\right],\left(a^{2}<1\right)\)
Find the sum of the series \(\frac{x^{2}}{1.2}-\frac{x^{3}}{2.3}+\frac{x^{4}}{3.4}-\ldots+(-1)^{n+1} \frac{x^{n+1}}{n(n+1)}+\ldots,|x|<1\)
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