Odd and even powers of sine

Use the substitution \(u=\sin x\) to show that \(\displaystyle\int\sin^nx\cos x\,\mathrm dx=\dfrac{\sin^{n+1}x}{n+1}+C\), where \(n\) is a positive integer.[2]

By writing \(\sin^3x=\sin x\left(1-\cos^2x\right)\), find \(\displaystyle\int\sin^3x\,\mathrm dx\).[3]

Explain why the method of (b) works for \(\displaystyle\int\sin^5x\,\mathrm dx\) but not for \(\displaystyle\int\sin^4x\,\mathrm dx\). Use the identity \(\sin^2x=\dfrac{1-\cos2x}{2}\) twice to find \(\displaystyle\int\sin^4x\,\mathrm dx\).[4]