What is the integral of e^{cos(pix)}sin(pix) ?

The integral of e^{cos(pix)}sin(pix) is -1/pi e^{cos(pix)}+C

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integral of e^{cos(pix)}sin(pix)

\int e^{c o s (π x)} sin (π x) d x

- \frac{1}{π} e^{c o s (π x)} + C

Solution steps

\int e^{c o s (π x)} sin (π x) d x

Apply u-substitution

: \int e^{c o s (u)} sin (u) \frac{1}{π} d u

= \int e^{c o s (u)} 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 e^{c o s (u)} sin (u) d u

Apply u-substitution

: \int - e^{v} d v

= \frac{1}{π} \cdot \int - e^{v} d v

Take the constant out:

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

= \frac{1}{π} (- \int e^{v} d v)

Use the common integral:

\int e^{v} d v = e^{v}

= \frac{1}{π} (- e^{v})

Substitute back

= \frac{1}{π} (- e^{c o s (π x)})

Simplify

= - \frac{1}{π} e^{c o s (π x)}

Add a constant to the solution

= - \frac{1}{π} e^{c o s (π x)} + C

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What is the integral of e^{cos(pix)}sin(pix) ?
The integral of e^{cos(pix)}sin(pix) is -1/pi e^{cos(pix)}+C