What is the integral of 2xsin(nx) ?

The integral of 2xsin(nx) is 2(-1/n xcos(nx)+1/(n^2)sin(nx))+C

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integral of 2xsin(nx)

\int 2 x sin (n x) d x

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

Solution steps

\int 2 x sin (n x) d x

Take the constant out:

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

= 2 \cdot \int x sin (n x) d x

Apply Integration By Parts

: - \frac{1}{n} x cos (n x) - \int - \frac{1}{n} cos (n x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of 2xsin(nx) ?
The integral of 2xsin(nx) is 2(-1/n xcos(nx)+1/(n^2)sin(nx))+C