Science:Math Exam Resources/Courses/MATH101 A/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 ${\sin }^2(x)+{\cos }^2(x)=1$:

$$\underset{0}{\overset{\pi /2}{\int }}{\sin }^4(x){\cos }^3(x)\mathrm{d}x=\underset{0}{\overset{\pi /2}{\int }}{\sin }^4(x)(1-{\sin }^2(x))\cos (x)\mathrm{d}x=\underset{0}{\overset{\pi /2}{\int }}({\sin }^{}(x)-{\sin }^{}(x))\cos (x)\mathrm{d}x.$$

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

$$\underset{0}{\overset{\pi /2}{\int }}{\sin }^4(x){\cos }^3(x)\mathrm{d}x=\underset{\sin (0)}{\overset{\sin (\pi /2)}{\int }}(u^4-u^6)\mathrm{d}u={[\frac{u^5}{5}-\frac{u^7}{7}]}_0^1=\frac{1}{5}-\frac{1}{7}-0+0=\frac{2}{35}.$$