What is the integral of-25sin(5x-pi/2) ?

The integral of-25sin(5x-pi/2) is 5sin(5x)+C

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integral of-25sin(5x-pi/2)

\int - 25 sin (5 x - \frac{π}{2}) d x

5 sin (5 x) + C

Solution steps

\int - 25 sin (5 x - \frac{π}{2}) d x

Take the constant out:

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

= - 25 \cdot \int sin (5 x - \frac{π}{2}) d x

Rewrite using trig identities

= - 25 \cdot \int - cos (5 x) d x

Take the constant out:

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

= - 25 (- \int cos (5 x) d x)

Apply u-substitution

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

= - 25 (- \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

= - 25 (- \frac{1}{5} \cdot \int cos (u) d u)

Use the common integral:

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

= - 25 (- \frac{1}{5} sin (u))

Substitute back

u = 5 x

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

Simplify

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

= 5 sin (5 x)

Add a constant to the solution

= 5 sin (5 x) + C

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What is the integral of-25sin(5x-pi/2) ?
The integral of-25sin(5x-pi/2) is 5sin(5x)+C