What is the integral of 5x^2sin(pix) ?

The integral of 5x^2sin(pix) is 5(-1/pi x^2cos(pix)+2/pi (1/pi xsin(pix)+1/(pi^2)cos(pix)))+C

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integral of 5x^2sin(pix)

\int 5 x^{2} sin (π x) d x

5 (- \frac{1}{π} x^{2} cos (π x) + \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x))) + C

Solution steps

\int 5 x^{2} sin (π x) d x

Take the constant out:

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

= 5 \cdot \int x^{2} sin (π x) d x

Apply Integration By Parts

: - \frac{1}{π} x^{2} cos (π x) - \int - \frac{2}{π} x cos (π x) d x

= 5 (- \frac{1}{π} x^{2} cos (π x) - \int - \frac{2}{π} x cos (π x) d x)

\int - \frac{2}{π} x cos (π x) d x = - \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x))

= 5 (- \frac{1}{π} x^{2} cos (π x) - (- \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x))))

Simplify

= 5 (- \frac{1}{π} x^{2} cos (π x) + \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x)))

Add a constant to the solution

= 5 (- \frac{1}{π} x^{2} cos (π x) + \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x))) + C

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\int 2 + 2 cos (x) d x

\int \frac{u ^{2}}{1 + u ^{2}} d u

What is the integral of 5x^2sin(pix) ?
The integral of 5x^2sin(pix) is 5(-1/pi x^2cos(pix)+2/pi (1/pi xsin(pix)+1/(pi^2)cos(pix)))+C