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प्रश्न
Evaluate the definite integral:
`int_0^1 (xe^x + sin (pix)/4)`
उत्तर
Let `I = int_0^1 [x e^x + sin ((pix)/4)] dx`
`= int_0^1 xe^x dx + int_0^1 sin (pix)/4 dx`
`|xe^x|_0^1 - int_0^1 (d/dx (x)* inte^x dx) dx - [(cos (pix)/4)/(pi/4)]_0^1`
`= |xe^x|_0^1 - int_0^1 e^x dx - 4/ pi [cos (pix)/4]_0^1`
`= [xe^x - e^x]_0^1 - 4/pi (cos pi/4 - cos 0)`
`= (e^1 - 0) - (e - e^0) - 4/pi (1/sqrt2 - 1)`
`= e - e + 1 - 4/ (pisqrt2) + 4/pi`
`= 1 + 4/ pi - (2sqrt2)/pi`
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