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Evaluate the integral using substitution

Use the method of substitution to evaluate the following integral:

    \[ \int \frac{\cos x \, dx}{\sin^3 x}. \]


    \[ u = \sin x \quad \implies \quad du = \cos x \, dx. \]

Then, we can evaluate the integral,

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

One comment

  1. Anonymous says:

    This can be solved by rewriting the initial problem as cot(x)csc^2(x), then setting u=cot(x). This method gives you a final answer of -cot^2(x)/2+C, which can be made to fit the answer in the textbook by using the trig identity cot^2(x)=csc^2(x)-1.

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