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Question
\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]
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Solution
\[\text{Let }I = \int_0^1 \left( x e^x + \cos \frac{\pi x}{4} \right) d x . Then, \]
\[I = \int_0^1 x e^x dx + \int_0^1 \cos\frac{\pi x}{4} dx\]
\[\text{Integrating first term by parts}\]
\[I = \left\{ \left[ x e^x \right]_0^1 - \int_0^1 1 e^x dx \right\} + \left[ \frac{\sin \frac{\pi x}{4}}{\frac{\pi}{4}} \right]_0^1 \]
\[ \Rightarrow I = \left[ x e^x \right]_0^1 - \left[ e^x \right]_0^1 + \left[ \frac{\sin \frac{\pi x}{4}}{\frac{\pi}{4}} \right]_0^1 \]
\[ \Rightarrow I = e - e + 1 + \frac{4}{\pi} \sin \frac{\pi}{4}\]
\[ \Rightarrow I = 1 + \frac{4}{\pi\sqrt{2}}\]
\[ \Rightarrow I = 1 + \frac{2\sqrt{2}}{\pi}\]
Notes
The answer given in the book has some error. The solution here is created according to the question given in the book.
