What is the integral of 10sin(5t) ?

The integral of 10sin(5t) is -2cos(5t)+C

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integral of 10sin(5t)

\int 10 sin (5 t) d t

- 2 cos (5 t) + C

Solution steps

\int 10 sin (5 t) d t

Take the constant out:

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

= 10 \cdot \int sin (5 t) d t

Apply u-substitution

= 10 \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

= 10 \cdot \frac{1}{5} \cdot \int sin (u) d u

Use the common integral:

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

= 10 \cdot \frac{1}{5} (- cos (u))

Substitute back

u = 5 t

= 10 \cdot \frac{1}{5} (- cos (5 t))

Simplify

10 \cdot \frac{1}{5} (- cos (5 t)) : - 2 cos (5 t)

= - 2 cos (5 t)

Add a constant to the solution

= - 2 cos (5 t) + C