Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 05/Solution 1

Following the hint, we wish to perform the substitution $u=\sin(x)$, but we must first rewrite the integrand by using the fundamental identity $\sin ^{2}(x)+\cos ^{2}(x)=1$:

$$\int _{0}^{\pi /2}\sin ^{4}(x)\cos ^{3}(x)\mathrm {d} x=\int _{0}^{\pi /2}\sin ^{4}(x)\left(1-\sin ^{2}(x)\right)\cos(x)\mathrm {d} x=\int _{0}^{\pi /2}\left(\sin ^{4}(x)-\sin ^{6}(x)\right)\cos(x)\mathrm {d} x.$$

We now perform the $u$-substitution above. Remember to change the integration bounds for the integral in terms of $u$:

$$\int _{0}^{\pi /2}\sin ^{4}(x)\cos ^{3}(x)\mathrm {d} x=\int _{\sin(0)}^{\sin(\pi /2)}(u^{4}-u^{6})\mathrm {d} u=\left[{\frac {u^{5}}{5}}-{\frac {u^{7}}{7}}\right]_{0}^{1}={\frac {1}{5}}-{\frac {1}{7}}-0+0={\frac {2}{35}}.$$