Question

\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]

 

Answer 1

\[\text{Let }I = \int_0^1 \left( x e^x + \cos \frac{\pi x}{4} \right) d x . Then, \]

\[I = \int_0^1 x e^x dx + \int_0^1 \cos\frac{\pi x}{4} dx\]

\[\text{Integrating first term by parts}\]

\[I = \left\{ \left[ x e^x \right]_0^1 - \int_0^1 1 e^x dx \right\} + \left[ \frac{\sin \frac{\pi x}{4}}{\frac{\pi}{4}} \right]_0^1 \]

\[ \Rightarrow I = \left[ x e^x \right]_0^1 - \left[ e^x \right]_0^1 + \left[ \frac{\sin \frac{\pi x}{4}}{\frac{\pi}{4}} \right]_0^1 \]

\[ \Rightarrow I = e - e + 1 + \frac{4}{\pi} \sin \frac{\pi}{4}\]

\[ \Rightarrow I = 1 + \frac{4}{\pi\sqrt{2}}\]

\[ \Rightarrow I = 1 + \frac{2\sqrt{2}}{\pi}\]