What is the integral of (pi+x)cos(nx) ?

The integral of (pi+x)cos(nx) is 1/n sin(nx)(pi+x)+1/(n^2)cos(nx)+C

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integral of (pi+x)cos(nx)

\int (π + x) cos (n x) d x

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

Solution steps

\int (π + x) cos (n x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (pi+x)cos(nx) ?
The integral of (pi+x)cos(nx) is 1/n sin(nx)(pi+x)+1/(n^2)cos(nx)+C