What is the integral of sin(nt) ?

The integral of sin(nt) is -1/n cos(nt)+C

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integral of sin(nt)

\int sin (n t) d t

- \frac{1}{n} cos (n t) + C

Solution steps

\int sin (n t) d t

Apply u-substitution

: \int \frac{sin ( u )}{n} d u

= \int \frac{sin ( u )}{n} d u

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{n} (- cos (u))

Substitute back

u = n t

= \frac{1}{n} (- cos (n t))

Simplify

= - \frac{1}{n} cos (n t)

Add a constant to the solution

= - \frac{1}{n} cos (n t) + C

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\int_{0}^{1} x (1 - x^{2}) d x

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\frac{d}{d x} (\frac{x ^{4} + 1}{x ^{2}})

\frac{d}{d x} (5 e^{- 4 x})