Evaluate the integral of cotx5 cscx2

We have to find the integral of \(\int \cot x^{5}. \csc x^{2}\)

Solution

\(\int \cot x^{5}. \csc x^{2}\)

Let us now substitute

cot x = t

If we differentiate both sides with respect to t we get,

\(\frac{\mathrm{d} }{\mathrm{d} x} cot x=\frac{dt}{dt}\)

Then we get,

\(-\csc^{2}x\frac{dx}{dt}= 1\)

On rearraging,

\(\csc^{2}x dx = -dt\)

On substituting these values in the given integral, we get

\(I=\int t^{5}dt\) \(I=\int t^{5}dt= \frac{t^{6}}{6} + C\)

On substituting the value of t = cot x we get

\(I=\frac{cot^{6}x}{6} + C\) \(\int \cot x^{5}. \csc x^{2}= \frac{cot^{6}x}{6} + C\)

Answer

\(\int \cot x^{5}. \csc x^{2}= \frac{cot^{6}x}{6} + C\)

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