Problem 3
Suppose \(\mathrm{f}\) and \(\mathrm{g}\) are continuous and \(f(a)=f(b)=0\) Prove \(\int_{a}^{b} f(x) g(x) d x=-\int_{a}^{b} f^{\prime}(x) G(x) d x\) where \(\mathrm{G}(\mathrm{x})=\int_{\mathrm{a}}^{\mathrm{x}} \mathrm{g}(\mathrm{t}) \mathrm{dt}\).
Problem 3
Find \(\int_{-1}^{2}[f(x)+2 g(x)] d x\) if \(\int_{-1}^{2} f(x) d x=5\) and \(\int_{-1}^{2} g(x) d x=-3\)
Problem 4
Evaluate the following limits: (i) \(\lim _{n \rightarrow x}\left[\tan \frac{\pi}{2 n} \tan \frac{2 \pi}{2 n} \tan \frac{3 \pi}{2 n} \ldots . \tan \frac{n \pi}{2 n}\right]^{1 / n}\) (ii) \(\lim _{n \rightarrow x}\left[\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{1 / 2}\left(1+\frac{3}{n}\right)^{1 / 3} \ldots\left(1+\frac{n}{n}\right)^{v_{n}}\right]\)
Problem 4
Find the following derivatives: (i) \(\frac{d}{d x} \int_{a}^{b} \sin \left(x^{2}\right) d x\) (ii) \(\frac{d}{d a} \int_{a}^{x^{2}} \sin \left(x^{2}\right) d x\) (iii) \(\frac{\mathrm{d}}{\mathrm{dx}} \int_{0}^{\mathrm{x}^{2}} \sqrt{1+\mathrm{x}^{2}} \mathrm{~d} \mathrm{x}\) (iv) \(\frac{d}{d x} \int_{x^{2}}^{x^{3}} \frac{d t}{\sqrt{1+t^{2}}}\) (v) \(\frac{d}{d x} \int_{x^{2}}^{x^{3}} \frac{d x}{\sqrt{1+x^{2}}}\) (vi) \(\frac{\mathrm{d}}{\mathrm{dx}} \int_{\mathrm{t}^{2}}^{\mathrm{x}^{3}} \frac{\mathrm{dt}}{\sqrt{\mathrm{x}^{2}+\mathrm{t}^{4}}}\)
Problem 4
If \(f^{\prime}\) is continuous on \([a, b]\), show that \(2 \int_{a}^{b} f(x) f^{\prime}(x) d x=[f(b)]^{2}-[f(a)]^{2}\)
Problem 4
If the function \(f\) is integrable in a closed interval containing \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) and \(\mathrm{d}\), prove that \(\int_{a}^{b} f(x) d x+\int_{b}^{c} f(x) d x+\int_{c}^{d} f(x) d x=\int_{a}^{d} f(x) d x\).
Problem 4
Evaluate \(\int_{0}^{1}\left(\sqrt[3]{1-x^{7}}-\sqrt[7]{1-x^{3}}\right) d x\)
Problem 4
Evaluate \(\int_{0}^{1} \frac{\tan ^{-1} \mathrm{ax}}{\mathrm{x} \sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}\)
Problem 4
A periodic function with period 1 is integrable over any finite interval. Also for two real numbers \(\mathrm{a}, \mathrm{b}\) and for two unequal non-zero postive integers \(\mathrm{m}\) and \(\mathrm{n}, \int_{a}^{a+1} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\int_{\mathrm{b}}^{b+\mathrm{m}} \mathrm{f}(\mathrm{x}) \mathrm{dx} .\) Calculate the value of \(\int_{\mathrm{m}}^{\mathrm{n}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\)
Problem 4
Evaluate the following integrals : (i) \(\int_{0}^{3 \pi / 2} \cos ^{4} 3 x \cdot \sin ^{2} 6 x d x\) (ii) \(\int_{0}^{1} x^{6} \sin ^{-1} x d x\) (iii) \(\int_{0}^{1} x^{3}(1-x)^{9 / 2} d x\) (iv) \(\int_{0}^{1} x^{4}(1-x)^{1 / 4} d x\)
