Evaluate the following definite integral: `int_0^1(x e^x+cos(pix)/4)dx`

Explanation

Take `I=int_{0}^{1}(x e^{x}+cos frac{pi x}{4}) d x. `

So , `I=int_{0}^{1} x e^{x} d x+int_{0}^{1} cos frac{pi x}{4} d x`

By using Integrating by parts on first term
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I={[x e^{x}]_{0}^{1}-int_{0}^{1} 1 e^{x} d x}+[frac{sin frac{pi x}{4}}{frac{pi}{4}}]_{0}^{1} `
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Rightarrow I=[x e^{x}]_{0}^{1}-[e^{x}]_{0}^{1}+[frac{sin frac{pi x}{4}}{frac{pi}{4}}]_{0}^{1} `
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Rightarrow I=e-e+1+frac{4}{pi} sin frac{pi}{4} `
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Rightarrow I=1+frac{4}{pi sqrt{2}} `
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