91Ó°ÊÓ

Question: Integrate the given function: \(\int \sin ^{m} x \cos ^{n} x\, dx\), where \(m\) is even and \(n\) is odd. Answer: To integrate the given function, follow these steps: 1. Observe that we have \(\sin ^{m} x \cos ^{n} x\) with \(m\) being even and \(n\) being odd. 2. Perform the u-substitution: let \(u=\sin x\) and \(du = \cos x\, dx\). This converts the integral to \(\int u^m \cos ^{n-1} x\, du\). 3. Use the Pythagorean identity to rewrite the remaining cosine term: \(\cos ^{n-1} x = (1 - u^2)^{\frac{n-1}{2}}\). Substitute this back into the integral, resulting in \(\int u^m (1 - u^2)^{\frac{n-1}{2}}\, du\). 4. Integrate the expression \(\int u^m (1 - u^2)^{\frac{n-1}{2}}\, du\) using appropriate integration techniques, such as integration by parts, partial fractions, or using a table of integrals. 5. Convert back to the original variable \(x\) by reversing the substitution \(u=\sin x\), and write down the solution as a function of \(x\) + constant (C).