Evaluate the following definite integral: `int_0^1(x e^(2x)+sin(pix)/2)dx`

Explanation

Take `I =int_{0}^{1}(x e^{2 x}+sin frac{pi x}{2}) d x.Then I=int_{0}^{1} x e^{2 x} d x+int_{0}^{1} sin frac{pi x}{2} d x `
by using Integrating first by parts


`I=[x frac{e^{2 x}}{2}]_{0}^{1}-int_{0}^{1} 1 frac{e^{2 x}}{2} d x+[-frac{cos frac{p i x}{2}}{frac{pi}{2}}]_{0}^{1} `
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Rightarrow I=[x frac{e^{2 x}}{2}]_{0}^{1}-[frac{e^{2 x}}{4}]_{0}^{1}-frac{2}{pi}[cos frac{p i x}{2}]_{0}^{1} `
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Rightarrow I=frac{e^{2}}{2}-frac{e^{2}}{4}+frac{1}{4}+frac{2}{pi} `
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Rightarrow I=frac{e^{2}}{4}+frac{1}{4}+frac{2}{pi}`