Evaluate the integral of cotx5 cscx2

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]

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