Evaluate the following definite integrals: _ 0 ^1 (xe^ 2x + 2 )dx - Vidyadip

Question

Evaluate the following definite integrals:$\int_{0}^\limits{1}\Big(\text{xe}^{2\text{x}}+\sin\frac{\pi\text{x}}{2}\Big)\text{dx}$

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Answer

Let $\text{I}=\int_{0}^\limits{1}\Big(\text{xe}^{2\text{x}}+\sin\frac{\pi\text{x}}{2}\Big)\text{dx}$ Then,$\text{I}=\int_{0}^\limits{1}\text{xe}^{2\text{x}}\text{ dx}+\int_{0}^\limits{1}\sin\frac{\pi\text{x}}{2}\text{ dx}$
Integrating first term by parts,$\text{I}=\Big[\text{x }\frac{\text{e}^{2\text{x}}}{2}\Big]^1_0-\int_{0}^\limits{1}1\frac{\text{e}^{2\text{x}}}{2}\text{ dx}+\bigg[-\frac{\cos\frac{\pi\text{x}}{2}}{\frac{\pi}{2}}\bigg]_0^1$
$\Rightarrow\text{I}=\Big[\text{x }\frac{\text{e}^{2\text{x}}}{2}\Big]^1_0-\Big[\frac{\text{e}^{2\text{x}}}{4}\Big]^1_0-\frac{2}{\pi}\Big[\cos\frac{\pi\text{x}}{2}\Big]^1_0$
$\Rightarrow\text{I}=\frac{\text{e}^{2}}{2}-\frac{\text{e}^{2}}{4}+\frac{1}{4}+\frac{2}{\pi}$
$\Rightarrow\text{I}=\frac{\text{e}^{2}}{4}+\frac{1}{4}+\frac{2}{\pi}$

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