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