Evaluate the following integrals:


Evaluate the following integrals:

$\int \operatorname{cosec}^{4} 3 x d x$


Let I $=\int \operatorname{cosec}^{4} 3 x d x$

$\Rightarrow I=\int \operatorname{cosec}^{2} 3 x \operatorname{cosec}^{2} 3 x d x$

$\Rightarrow I=\int\left(1+\cot ^{2} 3 x\right) \operatorname{cosec}^{2} 3 x d x$

$\Rightarrow I=\int\left(\operatorname{cosec}^{2} 3 x+\cot ^{2} 3 x \operatorname{cosec}^{2} 3 x\right) d x$

Let $\cot 3 x=t$, then

$\Rightarrow-3 \operatorname{cosec}^{2} 3 x d x=d t$

$\Rightarrow I=-\frac{1}{3} t-\frac{1}{3} \cdot \frac{1}{3} t^{3}+c$

$\Rightarrow I=-\frac{1}{3} \cot 3 x-\frac{1}{9} \cot ^{3} 3 x+c$

Therefore, $\int \operatorname{cosec}^{4} 3 x d x=-\frac{1}{3} \cot 3 x-\frac{1}{9} \cot ^{3} 3 x+c$

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