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

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

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integral of-5sin(5t)

\int - 5 sin (5 t) d t

cos (5 t) + C

Solution steps

\int - 5 sin (5 t) d t

Take the constant out:

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

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

Apply u-substitution

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

= - 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 t

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

Simplify

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

= cos (5 t)

Add a constant to the solution

= cos (5 t) + C

y^{3}

\int_{0}^{x} sin^{2} (t) d t

\frac{d x}{d y}, y^{2} - x = 0

f (x) = x^{4} + 8 x^{3} - 2 x - 2, at x = - 2

\int \frac{sec ^{2} ( x )}{1 + tan ( x )} d x