What is the integral of pi/2 x^2e^{pix} ?

The integral of pi/2 x^2e^{pix} is 1/2 e^{pix}x^2-1/pi e^{pix}x+1/(pi^2)e^{pix}+C

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integral of pi/2 x^2e^{pix}

\int \frac{π}{2} x^{2} e^{π x} d x

\frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x} + C

Solution steps

\int \frac{π}{2} x^{2} e^{π x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{π}{2} \cdot \int x^{2} e^{π x} d x

Apply Integration By Parts

: \frac{1}{π} e^{π x} x^{2} - \int \frac{2}{π} e^{π x} x d x

= \frac{π}{2} (\frac{1}{π} e^{π x} x^{2} - \int \frac{2}{π} e^{π x} x d x)

\int \frac{2}{π} e^{π x} x d x = \frac{2}{π} (\frac{1}{π} e^{π x} x - \frac{1}{π ^{2}} e^{π x})

= \frac{π}{2} (\frac{1}{π} e^{π x} x^{2} - \frac{2}{π} (\frac{1}{π} e^{π x} x - \frac{1}{π ^{2}} e^{π x}))

Simplify

\frac{π}{2} (\frac{1}{π} e^{π x} x^{2} - \frac{2}{π} (\frac{1}{π} e^{π x} x - \frac{1}{π ^{2}} e^{π x})) : \frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x}

= \frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x}

Add a constant to the solution

= \frac{1}{2} e^{π x} x^{2} - \frac{1}{π} e^{π x} x + \frac{1}{π ^{2}} e^{π x} + C

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What is the integral of pi/2 x^2e^{pix} ?
The integral of pi/2 x^2e^{pix} is 1/2 e^{pix}x^2-1/pi e^{pix}x+1/(pi^2)e^{pix}+C