Evaluate the definite integrals `int_0^1(x e^x+sin(pix)/4)dx`

Explanation

`int_0^1 (xe^x) +sin((pix)/4)dx`
Here, we will do integration by parts,
`int (xe^x) = xe^x-int e^xdx`
`int (xe^x) = xe^x-e^x+c`
Now, ` int sin((pix)/4)dx = -4/picos((pix)/4)+C`
`int_0^1 (xe^x) +sin((pix)/4)dx =[xe^x-e^x]_0^1+[-4/picos((pix)/4)]_0^1 `
` = e-e-(0-e^0)-4/pi(cospi/4-cos0)`
`=1-4/pi(1/sqrt2-1)`
`=1+4/pi((sqrt2-1)/sqrt2)`