What is the integral of picos(nt) ?

The integral of picos(nt) is pi/n sin(nt)+C

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integral of picos(nt)

\int π cos (n t) d t

\frac{π}{n} sin (n t) + C

Solution steps

\int π cos (n t) d t

Take the constant out:

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

= π \cdot \int cos (n t) d t

Apply u-substitution

= π \cdot \int \frac{cos ( u )}{n} d u

Take the constant out:

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

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

Use the common integral:

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

= π \frac{1}{n} sin (u)

Substitute back

u = n t

= π \frac{1}{n} sin (n t)

Simplify

π \frac{1}{n} sin (n t) : \frac{π}{n} sin (n t)

= \frac{π}{n} sin (n t)

Add a constant to the solution

= \frac{π}{n} sin (n t) + C