What is the integral of (2x+1)cos(4x) ?

The integral of (2x+1)cos(4x) is 1/4 sin(4x)(2x+1)+1/8 cos(4x)+C

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integral of (2x+1)cos(4x)

\int (2 x + 1) cos (4 x) d x

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

Solution steps

\int (2 x + 1) cos (4 x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (2x+1)cos(4x) ?
The integral of (2x+1)cos(4x) is 1/4 sin(4x)(2x+1)+1/8 cos(4x)+C