Question

Use integration by parts to verify the formula. (For Exercises \(57-60\), assume that \(n\) is a positive integer.) $$ \int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x $$

Short answer

The integration by parts formula for the given problem is verified as \(\int x^n \cos x \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx\).

Step-by-step solution

Selecting Parts

Let the parts be chosen such that \(u = x^n\) and \(dv = \cos x \, dx\). For the integration by parts, the fundamental formula is \(\int udv = uv - \int vdu\). Here \(n\) is a constant so its differentiation would reduce the power by one, while the integral of \(\cos x\) is \(\sin x\).

Differentiation and Integration

The next step involves differentiating and integrating the above chosen parts. Compute \(du\) by differentiating \(u=x^n\) and \(v\) by integrating \(dv=\cos x \, dx\). Thus, \(du = n \cdot x^{n-1} \, dx\) and \(v = \sin x\).

Using Formula of Integration by Parts

Using the formula of integration by parts \(\int udv = uv - \int vdu\). Substituting \(u\), \(v\), \(du\) and \(dv\) we get, \(\int x^n \cos x \, dx = x^n \sin x - \int \sin x \, n \cdot x^{n-1} \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx\). The formula is verified.