What is the integral of cos(pix) ?

The integral of cos(pix) is -i/p sin(pix)+C

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integral of cos(pix)

\int cos (p i x) d x

- \frac{i}{p} sin (p i x) + C

Solution steps

\int cos (p i x) d x

Apply u-substitution

: \int - \frac{i cos ( u )}{p} d u

= \int - \frac{i cos ( u )}{p} d u

Take the constant out:

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

= - \frac{i}{p} \cdot \int cos (u) d u

Use the common integral:

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

= - \frac{i}{p} sin (u)

Substitute back

u = p i x

= - \frac{i}{p} sin (p i x)

Add a constant to the solution

= - \frac{i}{p} sin (p i x) + C

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