Expert answers to "Evaluate the following integrals. ∫cot53xdx"

Daniaal Sanchez

Answered question

2021-10-20

Evaluate the following integrals.

\int \cot^{5} 3 x d x

AGRFTr

Skilled2021-10-21Added 95 answers

Evaluate:

\int \cot^{5} 3 x d x

Since

1 + \cot^{2} x = \csc^{2} x

so,

\int \cot^{5} 3 x d x = \frac{1}{3} \int \cot^{5} u d u

= \frac{1}{3} \int \cot^{3} u (\csc^{2} u - 1) d u

= \frac{1}{3} \int (\cot^{3} u \csc^{2} u - \cot^{3} u) d u

= \frac{1}{3} \int \cot^{3} u \csc^{2} u d u - \frac{1}{3} \int \cot^{3} u d u

= - \frac{1}{3} \int \cot^{3} u d (\cot u) d u - \frac{1}{3} \int \cot u (\csc^{2} u - 1) d u

= - \frac{\cot^{4} 3 x}{12} + \frac{\cot^{2} 3 x}{6} + \frac{1}{3} \ln | \sin 3 x | + C

Hence

\int \cot^{5} 3 x d x = - \frac{\cot^{4} 3 x}{12} + \frac{\cot^{2} 3 x}{6} + \frac{1}{3} \ln | \sin 3 x | + C

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