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Question Number 59509 by aliesam last updated on 11/May/19

Answered by MJS last updated on 11/May/19

$$\int \csc \pi xdx=\int \frac{dx}{\sin \pi x}=$$$$[t=\pi x\to dx=\frac{dt}{\pi }]$$$$=\frac{1}{\pi }\int \frac{dt}{\sin t}=$$$$[u=\tan \frac{t}{2}\to dt=2\frac{du}{u^2+1}]$$$$=\frac{1}{\pi }\int \frac{du}{u}=\frac{1}{\pi }\ln u=\frac{1}{\pi }\ln \tan \frac{t}{2}=$$$$=\frac{1}{\pi }\ln \tan \frac{\pi }{2}x+C$$$$\underset{1/6}{\stackrel{1/3}{\int }}\csc \pi xdx=\frac{1}{2\pi }\ln \frac{7+4\sqrt{3}}{3}\approx .244351$$

Answered by aliesam last updated on 11/May/19

$$ohiamsosorymyanswerismistakeyouarerightthankyou$$

Commented by MJS last updated on 11/May/19

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{welcome}.\mathrm{you}\mathrm{used}\mathrm{a}\mathrm{different}\mathrm{path}\mathrm{which}$$$$\mathrm{leads}\mathrm{to}\mathrm{the}\mathrm{same}\mathrm{result}$$

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