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Chapter 2: Problem 15
Find the interval in which \(F(x)=\int_{-1}^{x}\left(e^{t}-1\right)(2-t) d t, \quad(x>-1)\) is increasing.
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4\. Prove that (i) \(\frac{2 \pi}{13}<\int_{0}^{2 \pi} \frac{\mathrm{dx}}{10+3 \cos \mathrm{x}}<\frac{2 \pi}{7}\) (ii) \(0<\int_{0}^{\pi / 4} x \sqrt{\tan x}<\frac{\pi^{2}}{32}\) (iii) \(\frac{1}{2}<\int_{\pi / 4}^{\pi / 2} \frac{\sin \mathrm{x}}{\mathrm{x}} \mathrm{dx}<\frac{1}{\sqrt{2}}\) (iv) \(\left|\int_{1}^{4} \frac{\sin x}{x} d x\right| \leq \frac{3}{2}\).
Find \(\int_{0}^{2} f(x) d x\), where
\(f(x)=\left\\{\begin{array}{l}\frac{1}{\sqrt[4]{x^{3}}} \quad \text { for } 0
\leq x \leq 1 \\ \frac{1}{\sqrt[4]{(x-1)^{3}}} & \text { for }
1
Prove that \(\int_{0}^{2 \lambda} \frac{\sin x}{x} d x=\int_{0}^{i} \frac{\sin 2 y}{y} d y=\frac{\sin ^{2} \lambda}{\lambda}+\int_{0}^{i} \frac{\sin ^{2} x}{x^{2}} d x .\) Deduce that \(\int_{0}^{\infty} \frac{\sin x}{x} d x=\int_{0}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x\) (It may be assumed that the integrals are convergent)
Find \(\int_{0}^{2} f(x) d x\), where
\(f(x)=\left\\{\begin{array}{c}\frac{1}{\sqrt[4]{x^{3}}} \quad \text { for } 0
\leq x \leq 1 \\ \frac{1}{\sqrt[4]{(x-1)^{3}}}\end{array}\right.\) for \(1
Prove that, as \(n \rightarrow \infty, \int_{0}^{1} \cos n x \tan ^{-1} x d x \rightarrow 0\).
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