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Chapter 2: Problem 2
Suppose \(\int_{1}^{x} f(t) d t=x^{2}-2 x+1 .\) Find \(f(x)\)
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Suppose f is continuous, \(f(0)=0, f(1)=1, f^{\prime}(x)>0\), and \(\int_{0}^{1} f(x) d x=\frac{1}{3}\). Find the value of the integral \(\int_{0}^{1} \mathrm{f}^{-1}(\mathrm{y}) \mathrm{dy}\)
Evaluate the integrals (i) \(\int_{0}^{b} \frac{x d x}{(1+x)^{3}}\) (ii) \(\int_{0}^{b} \frac{x^{2} d x}{(1+x)^{4}}\) and show that they converge to finite limits as \(\mathrm{b} \rightarrow \infty\)
Evaluate \(\int_{0}^{1} \frac{\tan ^{-1} \mathrm{ax}}{\mathrm{x} \sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}\)
Evaluate the following integrals: (i) \(\int_{0}^{\pi / 2} \sin ^{5} x \cos ^{4} x d x\) (ii) \(\int_{0}^{\frac{\pi}{2}} \sin ^{7} x \cos ^{4} x d x\) (iii) \(\int_{0}^{\pi / 2} \sin ^{3} x \cos ^{5} x d x\) (iv) \(\int_{0}^{\pi} \sin ^{6} \frac{x}{2} \cos ^{8} \frac{x}{2} d x\)
Prove that \(\lim _{\omega \rightarrow \infty} \frac{e^{k m^{2} x^{2}}}{\int_{a}^{b} e^{k m^{2} x^{2}} d x}= \begin{cases}0 & \text { if } x0, \mathrm{k}>0, \mathrm{~b}>\mathrm{a}>0)\)
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