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Verify the given integral formula

Verify the formula using any method.

    \[ \int \frac{\cos^m x}{\sin^n x} \, dx = \frac{\cos^{m-1} x}{(m-n)\sin^{n-1} x} + \frac{m-1}{m-n} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx + C \]

for n \neq m.


Proof. First, we recall from a previous exercise (Section 5.10, Exercise #19) that we used integration by parts to establish the following formula,

    \[ \int \frac{\cos^{m+1} x}{\sin^{n+1} x} \, dx = -\frac{1}{n} \cdot \frac{\cos^m x}{\sin^n x} - \frac{m}{n} \int \frac{\cos^{m-1} x}{\sin^{n-1} x} \, dx. \]

Applying this to the current integral (with m in place of m+1 and n in place of n+1) and computing, we have

    \begin{align*}   \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{1}{n-1} \cdot \frac{\cos^{m-1} x}{\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x}{\sin^{n-2} x} \, dx \\[9pt]  &= -\frac{1}{n-1} \cdot \frac{\cos^{m-1} x}{\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x \sin^2 x}{\sin^n x} \, dx \\[9pt]  &= -\frac{1}{n-1} \cdot \frac{\cos^{m-1} x}{\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2}x (1 - \cos^2 x)}{\sin^n x} \, dx \\[9pt]  &= -\frac{\cos^{m-1} x}{(n-1)\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2}x}{\sin^n x} \, dx + \frac{m-1}{n-1} \int \frac{\cos^m x}{\sin^n x} \, dx. \end{align*}

Bringing the last integral \frac{m-1}{n-1} \int \frac{\cos^m x}{\sin^n x} \, dx back to the left, we have

    \begin{align*}  && \left( 1 - \frac{m-1}{n-1} \right) \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m-1} x}{(n-1)\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx \\[9pt]  \implies && \frac{n-m}{n-1} \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m-1} x}{(n-1)\sin^{n-1} x} - \frac{m-1}{n-1} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx \\[9pt]  \implies && \int \frac{\cos^m x}{\sin^n x} \, dx &= -\frac{\cos^{m-1} x}{(n-m)\sin^{n-1} x} - \frac{m-1}{n-m} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx \\[9pt]  \implies && \int \frac{\cos^m x}{\sin^n x} \, dx &= \frac{\cos^{m-1} x}{(m-n) \sin^{n-1} x} + \frac{m-1}{m-n} \int \frac{\cos^{m-2} x}{\sin^n x} \, dx. \qquad \blacksquare \end{align*}

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