What is the integral of 5sin(5x-3) ?

The integral of 5sin(5x-3) is -cos(5x-3)+C

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integral of 5sin(5x-3)

\int 5 sin (5 x - 3) d x

- cos (5 x - 3) + C

Solution steps

\int 5 sin (5 x - 3) d x

Take the constant out:

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

= 5 \cdot \int sin (5 x - 3) d x

Apply u-substitution

= 5 \cdot \int sin (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 sin (u) d u

Use the common integral:

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

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

Substitute back

u = 5 x - 3

= 5 \cdot \frac{1}{5} (- cos (5 x - 3))

Simplify

5 \cdot \frac{1}{5} (- cos (5 x - 3)) : - cos (5 x - 3)

= - cos (5 x - 3)

Add a constant to the solution

= - cos (5 x - 3) + C