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Question: Prove that $\int_{-\pi}^{\pi} x^{12} \sin ^{9} x d x=0$ Solution:...
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Question: Prove that $\int_{-\pi}^{\pi}\left(\sin ^{75} x+x^{125}\right) d x=0$ Solution:...
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Question: Prove that $\int_{-a}^{a} x^{3} \sqrt{a^{2}-x^{2}} d x=0$ Solution:...
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Question: Prove that $\int_{0}^{1} \frac{\log (1+x)}{\left(1+x^{2}\right)} d x=\frac{\pi}{8}(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{1} \frac{\log x}{\sqrt{1-x^{2}}} d x=-\frac{\pi}{2}(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{1}\left(\frac{\sin ^{-1} x}{x}\right) d x=\frac{\pi}{2}(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{\pi / 2} x \cot x d x=\frac{\pi}{4}(\log 2)$ Solution:...
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Question: Prove that $\int_{1}^{4} \frac{\sqrt{x}}{(\sqrt{5-x}+\sqrt{x})} d x=\frac{3}{2}$ Solution:...
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Question: Prove that $\int_{\alpha / 4}^{3 \alpha / 4} \frac{\sqrt{x}}{(\sqrt{a-x}+\sqrt{x})} d x=\frac{a}{4}$ Solution:...
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Question: Prove that $\int_{\pi / 4}^{3 \pi / 4} \frac{x}{(1+\sin x)} d x=\pi(\sqrt{2}-1)$ Solution:...
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Question: Prove that $\int_{\pi / 4}^{3 \pi / 4} \frac{d x}{(1+\cos x)}=2$ Solution:...
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Question: Prove that $\int_{\pi / 6}^{\pi / 3} \frac{1}{(1+\sqrt{\tan x})} d x=\frac{\pi}{12}$ Solution:...
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Question: Prove that $\int_{\pi / 8}^{3 \pi / 8} \frac{\cos x}{(\cos x+\sin x)} d x=\frac{\pi}{8}$ Solution:...
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Question: Prove that $\int_{0}^{\pi / 2} \log (\tan x+\cot x) d x=\pi(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{\pi} \log (1+\cos x) d x=-\pi(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{\pi} x \log (\sin x) d x=-\frac{\pi^{2}}{2}(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{\pi / 2} \log (\sin 2 x) d x=-\frac{\pi}{2}(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{\pi / 2}(\sin x-\cos x) \log (\sin x+\cos x) d x=0$ Solution:...
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Question: Prove that $\int_{0}^{\pi} \sin ^{2 m} x \cos ^{2 m+1} x d x=0$, where $m$ is a positive integer Solution: $y=0$...
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Question: Prove that $\int_{0}^{\pi} \sin ^{2} x \cos ^{3} x d x=0$ Solution:...
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Question: $\int_{0}^{a} \frac{\sqrt{x}}{(\sqrt{x}+\sqrt{a-x})} d x=\frac{\pi}{4}$ Solution:...
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Question: Prove that $\int_{0}^{a} \frac{d x}{x+\sqrt{a^{2}-x^{2}}}=\frac{\pi}{4}$ Solution:...
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Question: Prove that $\int_{0}^{\infty} \frac{x}{(1+x)\left(1+x^{2}\right)} d x=\frac{\pi}{4}$ Solution:...
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Question: Prove that $\int_{0}^{\pi / 2}(2 \log \cos x-\log \sin 2 x) d x=-\frac{\pi}{4}(\log 2)$ Solution:...
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Question: Prove that $\int_{0}^{\pi} \frac{x}{\left(1+\sin ^{2} x\right)} d x=\frac{\pi^{2}}{2 \sqrt{2}}$ Solution:...
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