What is the integral of-sin(pix) ?

The integral of-sin(pix) is 1/pi cos(pix)+C

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integral of-sin(pix)

\int - sin (π x) d x

\frac{1}{π} cos (π x) + C

Solution steps

\int - sin (π x) d x

Take the constant out:

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

= - \int sin (π x) d x

Apply u-substitution

: \int sin (u) \frac{1}{π} d u

= - \int sin (u) \frac{1}{π} d u

Take the constant out:

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

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

Use the common integral:

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

= - \frac{1}{π} (- cos (u))

Substitute back

u = π x

= - \frac{1}{π} (- cos (π x))

Simplify

= \frac{1}{π} cos (π x)

Add a constant to the solution

= \frac{1}{π} cos (π x) + C

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