Evaluate the following definite integral: `int_0^(2pi)e^(x//2)sin(x/2+pi/4)dx`

Explanation

` I=int_{0}^{2 pi} e^{frac{x}{2}} sin (frac{x}{2}+frac{pi}{4}) d x`

Applying Integrating by parts
`
I=[-2 e^{frac{x}{2}} cos (frac{x}{2}+frac{pi}{4})]_{0}^{2 pi}-int_{0}^{2 pi}-frac{2}{2} e^{frac{x}{2}} cos (frac{x}{2}+frac{pi}{4}) d x`

Again, Applying integrating second term by parts
`

Rightarrow I=[-2 e^{frac{pi}{2}} cos (frac{x}{2}+frac{pi}{4})]_{0}^{2 pi}+{[2 e^{frac{x}{2}} sin (frac{x}{2}+frac{pi}{4})]_{0}^{2 pi}-int_{0}^{2 pi} frac{2}{2} e^{frac{x}{2}} sin (frac{x}{2}+frac{pi}{4}) d x} `
`
Rightarrow I=[-2 e^{frac{x}{2}} cos (frac{x}{2}+frac{pi}{4})]_{0}^{2 pi}+[2 e^{frac{x}{2}} sin (frac{x}{2}+frac{pi}{4})]_{0}^{2 pi}-I `
`
Rightarrow 2 I=frac{2}{sqrt{2}} e^{pi}+frac{2}{sqrt{2}}-frac{2}{sqrt{2}} e^{pi}-frac{2}{sqrt{2}}=0 `
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