What is the integral of cos(4x-pi/4) ?

The integral of cos(4x-pi/4) is 1/(4sqrt(2))sin(4x)-1/(4sqrt(2))cos(4x)+C

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integral of cos(4x-pi/4)

\int cos (4 x - \frac{π}{4}) d x

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

Solution steps

\int cos (4 x - \frac{π}{4}) d x

Rewrite using trig identities

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

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

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