What is the integral of e^{pix} ?

The integral of e^{pix} is 1/pi e^{pix}+C

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integral of e^{pix}

\int e^{π x} d x

\frac{1}{π} e^{π x} + C

Solution steps

\int e^{π x} d x

Apply u-substitution

: \int e^{u} \frac{1}{π} d u

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

Use the common integral:

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

= \frac{1}{π} e^{u}

Substitute back

u = π x

= \frac{1}{π} e^{π x}

Add a constant to the solution

= \frac{1}{π} e^{π x} + C

\int e^{2 x} 1 - e^{2 x} d x

y^{^{''}} + 2 y^{^{'}} + 2 y = cos (2 x)

\frac{d}{d x} (\frac{x}{y ^{2}})

\frac{\partial}{\partial x} (sin (- 2 x) cos (4 y))

\frac{d}{d x} (x^{2} + a \cdot x + b)