What is the integral of e^{at} ?

The integral of e^{at} is 1/a e^{at}+C

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integral of e^{at}

\int e^{a t} d t

\frac{1}{a} e^{a t} + C

Solution steps

\int e^{a t} d t

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{a} \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{a} e^{u}

Substitute back

u = a t

= \frac{1}{a} e^{a t}

Add a constant to the solution

= \frac{1}{a} e^{a t} + C

\frac{\partial}{\partial x} (\frac{1}{4} 2^{x} y^{3})

\int \frac{1}{16 + x ^{2}} d x

\frac{\partial}{\partial y} (8 x y - \frac{1}{4} (x + y)^{4})

x \to x lim (sin (x))

a re a y = x^{2}, y = 9 x, y = - 1