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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int x^{n-1} \sin (x^n) \, dx. \]

Let

    \[ u = x^n \quad \implies \quad du = nx^{n-1} \, dx. \]

Then, we can evaluate the integral,

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

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