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

The integral of sin(5x)5x) is -dxcos(5x)+1/5 dsin(5x)+C

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integral of sin(5x)5x)

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

- d x cos (5 x) + \frac{1}{5} d sin (5 x) + C

Solution steps

\int sin (5 x) d \cdot 5 x d x

Take the constant out:

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

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

Apply Integration By Parts

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

\int - \frac{1}{5} cos (5 x) d x = - \frac{1}{25} sin (5 x)

= d \cdot 5 (- \frac{1}{5} x cos (5 x) - (- \frac{1}{25} sin (5 x)))

Simplify

d \cdot 5 (- \frac{1}{5} x cos (5 x) - (- \frac{1}{25} sin (5 x))) : - d x cos (5 x) + \frac{1}{5} d sin (5 x)

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

Add a constant to the solution

= - d x cos (5 x) + \frac{1}{5} d sin (5 x) + C

What is the integral of sin(5x)5x) ?
The integral of sin(5x)5x) is -dxcos(5x)+1/5 dsin(5x)+C