Integer 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, then the value of 8(\alpha+\beta) equals \qquad .

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.

let \mathrm{I}_{1}=\int \operatorname{cosec}^{3} \mathrm{x} d \mathrm{dx}=\int \operatorname{cosec} x \cdot \operatorname{cosec}^{2} \mathrm{x} \mathrm{dx}=-\operatorname{cosec} x \cot \mathrm{x}-\int \cot ^{2} \mathrm{x} \operatorname{cosec} \mathrm{xdx}

I_{1}=-\operatorname{cosec} x \cot x-\int\left(\operatorname{cosec}^{2} x-1\right) \operatorname{cosec} x d x=-\operatorname{cosec} x \cot x-\int \operatorname{cosec}^{3} x d x+\int \operatorname{cosec} x d x

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