What is the integral of 3cos(3x-1) ?

The integral of 3cos(3x-1) is sin(3x-1)+C

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integral of 3cos(3x-1)

\int 3 cos (3 x - 1) d x

sin (3 x - 1) + C

Solution steps

\int 3 cos (3 x - 1) d x

Take the constant out:

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

= 3 \cdot \int cos (3 x - 1) d x

Apply u-substitution

= 3 \cdot \int cos (u) \frac{1}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= 3 \cdot \frac{1}{3} sin (u)

Substitute back

u = 3 x - 1

= 3 \cdot \frac{1}{3} sin (3 x - 1)

Simplify

3 \cdot \frac{1}{3} sin (3 x - 1) : sin (3 x - 1)

= sin (3 x - 1)

Add a constant to the solution

= sin (3 x - 1) + C