If n is a natural number, then int frac{sin^n | vedclass

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(D) We have the integral $I = \int \frac{\sin^n x}{\cos^{n+2} x} dx$.Rewrite the integrand as $I = \int \left( \frac{\sin x}{\cos x} \right)^n \cdot \

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$\frac{	an^{n+1} x}{n+1} + C$

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(D) We have the integral $I = \int \frac{\sin^n x}{\cos^{n+2} x} dx$.Rewrite the integrand as $I = \int \left( \frac{\sin x}{\cos x} ight)^n \cdot \frac{1}{\cos^2 x} dx$.Since $\frac{\sin x}{\cos x} = 	an x$ and $\frac{1}{\cos^2 x} = \sec^2 x$, we get $I = \int 	an^n x \sec^2 x dx$.Let $t = 	an x$, then $dt = \sec^2 x dx$.Substituting these into the integral, we get $I = \int t^n dt$.Using the power rule for integration, $I = \frac{t^{n+1}}{n+1} + C$.Substituting $t = 	an x$ back, we get $I = \frac{	an^{n+1} x}{n+1} + C$.

If $n$ is a natural number, then $\int \frac{\sin^n x}{\cos^{n+2} x} dx =$

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