We have to find the integral of [latex]\int \cot x^{5}. \csc x^{2}[/latex]
Solution
[latex]\int \cot x^{5}. \csc x^{2}[/latex]Let us now substitute
cot x = t
If we differentiate both sides with respect to t we get,
[latex]\frac{\mathrm{d} }{\mathrm{d} x} cot x=\frac{dt}{dt}[/latex]Then we get,
[latex]-\csc^{2}x\frac{dx}{dt}= 1[/latex]On rearraging,
[latex]\csc^{2}x dx = -dt[/latex]On substituting these values in the given integral, we get
[latex]I=\int t^{5}dt[/latex] [latex]I=\int t^{5}dt= \frac{t^{6}}{6} + C[/latex]On substituting the value of t = cot x we get
[latex]I=\frac{cot^{6}x}{6} + C[/latex] [latex]\int \cot x^{5}. \csc x^{2}= \frac{cot^{6}x}{6} + C[/latex]Answer
[latex]\int \cot x^{5}. \csc x^{2}= \frac{cot^{6}x}{6} + C[/latex]
