What is the integral of 2cos(2x-5) ?

The integral of 2cos(2x-5) is sin(2x-5)+C

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integral of 2cos(2x-5)

\int 2 cos (2 x - 5) d x

sin (2 x - 5) + C

Solution steps

\int 2 cos (2 x - 5) d x

Take the constant out:

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

= 2 \cdot \int cos (2 x - 5) d x

Rewrite using trig identities

= 2 \cdot \int cos (2 x) cos (5) + sin (2 x) sin (5) 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

= 2 (\int cos (2 x) cos (5) d x + \int sin (2 x) sin (5) d x)

\int cos (2 x) cos (5) d x = cos (5) \frac{1}{2} sin (2 x)

\int sin (2 x) sin (5) d x = - \frac{1}{2} sin (5) cos (2 x)

= 2 (cos (5) \frac{1}{2} sin (2 x) - \frac{1}{2} sin (5) cos (2 x))

Simplify

2 (cos (5) \frac{1}{2} sin (2 x) - \frac{1}{2} sin (5) cos (2 x)) : sin (2 x - 5)

= sin (2 x - 5)

Add a constant to the solution

= sin (2 x - 5) + C

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