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

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

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integral of 5cos(5x)

\int 5 cos (5 x) d x

sin (5 x) + C

Solution steps

\int 5 cos (5 x) d x

Take the constant out:

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

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

Apply u-substitution

: \int cos (u) \frac{1}{5} d u

= 5 \cdot \int cos (u) \frac{1}{5} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int cos (u) d u

Use the common integral:

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

= 5 \cdot \frac{1}{5} sin (u)

Substitute back

u = 5 x

= 5 \cdot \frac{1}{5} sin (5 x)

Simplify

5 \cdot \frac{1}{5} sin (5 x) : sin (5 x)

= sin (5 x)

Add a constant to the solution

= sin (5 x) + C

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