What is the integral of xcos^2(pix) ?

The integral of xcos^2(pix) is 1/2 x^2+1/(4pi)xsin(2pix)-(x^2)/4+1/(8pi^2)cos(2pix)+C

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integral of xcos^2(pix)

\int x cos^{2} (π x) d x

\frac{1}{2} x^{2} + \frac{1}{4 π} x sin (2 π x) - \frac{x ^{2}}{4} + \frac{1}{8 π ^{2}} cos (2 π x) + C

Solution steps

\int x cos^{2} (π x) d x

Apply Integration By Parts

: x (\frac{1}{2} x + \frac{1}{4 π} sin (2 π x)) - \int \frac{1}{2} x + \frac{1}{4 π} sin (2 π x) d x

= x (\frac{1}{2} x + \frac{1}{4 π} sin (2 π x)) - \int \frac{1}{2} x + \frac{1}{4 π} sin (2 π x) d x

\int \frac{1}{2} x + \frac{1}{4 π} sin (2 π x) d x = \frac{x ^{2}}{4} - \frac{1}{8 π ^{2}} cos (2 π x)

= x (\frac{1}{2} x + \frac{1}{4 π} sin (2 π x)) - (\frac{x ^{2}}{4} - \frac{1}{8 π ^{2}} cos (2 π x))

Simplify

x (\frac{1}{2} x + \frac{1}{4 π} sin (2 π x)) - (\frac{x ^{2}}{4} - \frac{1}{8 π ^{2}} cos (2 π x)) : \frac{1}{2} x^{2} + \frac{1}{4 π} x sin (2 π x) - \frac{x ^{2}}{4} + \frac{1}{8 π ^{2}} cos (2 π x)

= \frac{1}{2} x^{2} + \frac{1}{4 π} x sin (2 π x) - \frac{x ^{2}}{4} + \frac{1}{8 π ^{2}} cos (2 π x)

Add a constant to the solution

= \frac{1}{2} x^{2} + \frac{1}{4 π} x sin (2 π x) - \frac{x ^{2}}{4} + \frac{1}{8 π ^{2}} cos (2 π x) + C

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What is the integral of xcos^2(pix) ?
The integral of xcos^2(pix) is 1/2 x^2+1/(4pi)xsin(2pix)-(x^2)/4+1/(8pi^2)cos(2pix)+C