Indefinite Integration Question 1

Byshahkb4

If \int \operatorname{cosec}^{5} x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^{2} x+\frac{3}{2}\right)+\beta \log _{e}\left|\tan \frac{x}{2}\right|+C where \alpha, \beta \in R and C is the constant of integration,

Sol.(1) \int \operatorname{cosec}^{3} x \cdot \operatorname{cosec}^{2} x d x=1
By applying integration by parts
I=-\cot x \operatorname{cosec}^{3} x+\int \cot x\left(-3 \operatorname{cosec}^{2} x \cot x \operatorname{cosec} x\right) d x
I=-\cot x \operatorname{cosec}^{3} x-3 \int \operatorname{cosec}^{3} x\left(\operatorname{cosec}^{2} x-1\right) d x
I=-\cot x \operatorname{cosec}^{3} x-31+3 \int \operatorname{cosec}^{3} x d x
4 I=-\cot x \operatorname{cosec}^{3} x+3 \int \operatorname{cosec}^{3} x d x.

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