If f(x) = int operatorname{cosec}^5 x , dx,th | vedclass

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Question

(C) We use the reduction formula for $\int \operatorname{cosec}^n x \, dx = -\frac{\operatorname{cosec}^{n-2} x \cot x}{n-1} + \frac{n-2}{n-1} \int \o

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$-\frac{1}{8}[7 \sqrt{2} + 3 \log (\sqrt{2} + 1)] + c$

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(C) We use the reduction formula for $\int \operatorname{cosec}^n x \, dx = -\frac{\operatorname{cosec}^{n-2} x \cot x}{n-1} + \frac{n-2}{n-1} \int \operatorname{cosec}^{n-2} x \, dx$. For $n=5$,$\int \operatorname{cosec}^5 x \, dx = -\frac{\operatorname{cosec}^3 x \cot x}{4} + \frac{3}{4} \int \operatorname{cosec}^3 x \, dx$. For $n=3$,$\int \operatorname{cosec}^3 x \, dx = -\frac{\operatorname{cosec} x \cot x}{2} + \frac{1}{2} \int \operatorname{cosec} x \, dx = -\frac{\operatorname{cosec} x \cot x}{2} + \frac{1}{2} \log |\operatorname{cosec} x - \cot x|$. Substituting this back,$f(x) = -\frac{\operatorname{cosec}^3 x \cot x}{4} + \frac{3}{4} \left( -\frac{\operatorname{cosec} x \cot x}{2} + \frac{1}{2} \log |\operatorname{cosec} x - \cot x| \right) + c$. $f(x) = -\frac{1}{4} \operatorname{cosec}^3 x \cot x - \frac{3}{8} \operatorname{cosec} x \cot x + \frac{3}{8} \log |\operatorname{cosec} x - \cot x| + c$. At $x = \frac{\pi}{4}$,$\operatorname{cosec} x = \sqrt{2}$ and $\cot x = 1$. $f\left(\frac{\pi}{4}\right) = -\frac{1}{4} (\sqrt{2})^3 (1) - \frac{3}{8} (\sqrt{2}) (1) + \frac{3}{8} \log |\sqrt{2} - 1| + c$. $f\left(\frac{\pi}{4}\right) = -\frac{2\sqrt{2}}{4} - \frac{3\sqrt{2}}{8} + \frac{3}{8} \log (\sqrt{2} - 1) + c$. $f\left(\frac{\pi}{4}\right) = -\frac{4\sqrt{2} + 3\sqrt{2}}{8} - \frac{3}{8} \log (\sqrt{2} + 1) + c = -\frac{1}{8} [7\sqrt{2} + 3 \log (\sqrt{2} + 1)] + c$.

If $f(x) = \int \operatorname{cosec}^5 x \, dx$,then $f\left(\frac{\pi}{4}\right) = $

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