What is the integral of piysin(pixy) ?

The integral of piysin(pixy) is -cos(piyx)+C

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integral of piysin(pixy)

\int π y sin (π x y) d x

- cos (π y x) + C

Solution steps

\int π y sin (π x y) d x

Take the constant out:

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

= y π \cdot \int sin (y x π) d x

Apply u-substitution

: \int \frac{sin ( u )}{π y} d u

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

Take the constant out:

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

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

Use the common integral:

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

= y π \frac{1}{π y} (- cos (u))

Substitute back

u = y x π

= y π \frac{1}{π y} (- cos (y x π))

Simplify

y π \frac{1}{π y} (- cos (y x π)) : - cos (π y x)

= - cos (π y x)

Add a constant to the solution

= - cos (π y x) + C

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What is the integral of piysin(pixy) ?
The integral of piysin(pixy) is -cos(piyx)+C