What is the integral of sin(x/2)cos(nx) ?

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

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integral of sin(x/2)cos(nx)

\int sin (\frac{x}{2}) cos (n x) d x

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

Solution steps

\int sin (\frac{x}{2}) cos (n x) d x

Apply Integration By Parts

: - 2 cos (n x) cos (\frac{x}{2}) - \int 2 n sin (n x) cos (\frac{x}{2}) d x

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

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

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

Add a constant to the solution

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

\int ln (\frac{x}{1 - x}) d x

\int sin^{4} (\frac{x}{2}) cos (\frac{x}{2}) d x

\int (tan (x) sec (x)) d x

\int (sin^{2} (3 x)) (cos (3 x)) d x

\int \frac{cos ( y )}{x + 3} d x

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