Problem 63 - Essential Calculus

Question

In your own words, describe how you would integrate \(\int \sin ^{m} x \cos ^{n} x d x\) for each condition. (a) \(m\) is positive and odd. (b) \(n\) is positive and odd. (c) \(m\) and \(n\) are both positive and even.

Step-by-Step Solution

Answer

For case (a), use the identity \(\sin^2 x = 1 - \cos^2 x\) and substitute \(u = \cos x\). For case (b), use the identity \(\cos^2 x = 1 - \sin^2 x\) and substitute \(u = \sin x\). For case (c), use power-reducing formulas into basic trigonometric integrals.

Step 1: Case (a): m is positive and odd

This can be approached by saving one factor of \(\sin x\), then using the identity \(\sin^2 x = 1 - \cos^2 x\). Replace every other factor of \(\sin x\) in the integrand by \(1 - \cos^2 x\) and then make a substitution by letting \(u = \cos x\). The differential \(du\) will be \(-\sin x\, dx\). For the integral, the saved factor of \(\sin x\) allows the \(dx\) to be replaced by \(-du\).

Step 2: Case (b): n is positive and odd

Similar to Case (a), here one factor of \(\cos x\) should be saved. The identity \(\cos^2 x = 1 - \sin^2 x\) can be employed, replacing every other factor of \(\cos x\) in the integrand by \(1 - \sin^2 x\). Afterwards, the substitution of \(u = \sin x\) can be used, with \(du = \cos x\, dx\). The saved factor of \(\cos x\) then lets the \(dx\) be replaced by \(du\).

Step 3: Case (c): m and n are both positive and even

In this situation, use the power-reducing formulas \(\cos^2 x = (1 + \cos 2x)/2\) and \(\sin^2 x = (1 - \cos 2x)/2\) to reduce the powers of sine and cosine. The integral will turn into multiples of basic trigonometric integrals.