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Use integration by parts to prove a given integral formula

by RoRi

Using integration by parts prove the validity of the formula:

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

Using this solution evaluate the following integrals

    \[ \int \cot^2 x \, dx \qquad \text{and} \qquad \int \cot^4 x \, dx. \]

Proof. To use integration by parts define the following:

    \begin{align*} u &= \cos^m x & \implies && du &= -m \cos^{m-1} x \sin x \, dx \\ dv &= \frac{\cos x}{\sin^{n+1} x} \, dx & \implies && v &= -\frac{1}{n} \cdot \frac{1}{\sin^n x}. \end{align*}

Where the formula for v followed by making the substitution t = \sin x, dt = \cos x \, dx to evaluate

    \begin{align*} \int \frac{\cos x}{\sin^{n+1} x} \, dx &= \int \frac{1}{t^{n+1}} \, dt \\ &= -\frac{1}{n} t^{-n} \\ &= -\frac{1}{n} \cdot \frac{1}{\sin^n x}. \end{align*}

Therefore,

    \begin{align*} \int \frac{\cos^{m+1} x}{\sin^{n+1} x} \, dx &= \int u \, dv \\ &= uv - \int v \, du \\ &= -\frac{1}{n} \cdot \frac{\cos^m x}{\sin^n x} - \frac{m}{n} \int \frac{\cos^{m-1} x \sin x}{\sin^n 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. \qquad \blacksquare \end{align*}

We can then use this solution to evaluate \cot^2 x and \cot^4 x. For \cot^2 x we use the formula with m+1 = n+1 = 2,

    \begin{align*} \int \cot^2 x \, dx &= \int \frac{\cos^2 x}{\sin^2 x} \, dx \\ &= -\frac{\cos x}{\sin x} - \int dx \\ &= -\cot x - x + C. \end{align*}

For \cot^4 x we use the formula with m+1=n+1=4,

    \begin{align*} \int \cot^4 x \, dx &= \int \frac{\cos^4 x}{\sin^4 x} \, dx \\ &= -\frac{1}{3} \frac{\cos^3 x}{\sin^3 x} - \int \cot^2 x \, dx \\ &= -\frac{1}{3} \cot^3 x + \cot x + x + C. \end{align*}

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