Question

The value of $\underset{0}{\overset{1}{\int }}(x{e}^x+\sin \left(\frac{\pi x}{4}\right))dx$ is:

• A. $1+\frac{4}{\pi }-\frac{2\sqrt{2}}{\pi }$
• B. $1+\frac{4}{\pi }-\frac{\sqrt{3}}{\pi }$
• C. $1-\frac{4}{\pi }+\frac{2\sqrt{2}}{\pi }$
• D. $1-\frac{4}{\pi }+\frac{\sqrt{2}}{\pi }$

Answer 1

The correct option is A $1+\frac{4}{\pi }-\frac{2\sqrt{2}}{\pi }$

Let $I=\underset{0}{\overset{1}{\int }}(x{e}^x+\sin \left(\frac{\pi x}{4}\right))dx$
$=\underset{0}{\overset{1}{\int }}x{e}^x+\underset{0}{\overset{1}{\int }}\sin \left(\frac{\pi x}{4}\right)dx$
$={[x{e}^x-e^x]}_0^1+{[-\frac{\cos \frac{\pi x}{4}}{\frac{\pi }{4}}]}_0^1$
$=[e-e-0+1]-\frac{4}{\pi }[\cos \left(\frac{\pi }{4}\right)-\cos (0\left)\right]=1+\frac{4}{\pi }[1-\frac{1}{\sqrt{2}}]$
$=1+\frac{4}{\pi }-\frac{2\sqrt{2}}{\pi }$