Integrating exponential functions

Standard integrals of e^x and e^(ax+b), definite integrals, finding a function from its derivative, the reverse chain rule with f'(x)e^f(x), and finding an integral from a given derivative.

Standard forms

Standard integrals

$$\int e^{ax + b}\,dx = \frac1a e^{ax + b} + C \qquad \int f'(x)\,e^{f(x)}\,dx = e^{f(x)} + C$$

Example 1●

Find $\displaystyle\int \big(e^{2x} + 3\big)\,dx$.

$\tfrac12 e^{2x} + 3x + C$

Example 2●●

Evaluate (a) $\displaystyle\int_0^1 e^{2x}\,dx$ and (b) $\displaystyle\int_0^{\ln 3} e^x\,dx$.

$\Big[\tfrac12 e^{2x}\Big]_0^1 = \dfrac{e^2 - 1}{2}$

$\Big[e^x\Big]_0^{\ln 3} = 3 - 1 = 2$, since $e^{\ln 3} = 3$.

The reverse chain rule

Example 3●●

Evaluate $\displaystyle\int_0^1 xe^{x^2}\,dx$.

The derivative of $x^2$ is $2x$, so write it as $\tfrac12\displaystyle\int_0^1 2xe^{x^2}\,dx$.

$\tfrac12\Big[e^{x^2}\Big]_0^1 = \dfrac{e - 1}{2}$

Using a known value

Example 4●●

$f'(x) = e^{2x}$ and $f(0) = 2$. Find $f(x)$.

$f(x) = \tfrac12 e^{2x} + C$, and $f(0) = \tfrac12 + C = 2$, so $C = \tfrac32$.

$f(x) = \tfrac12 e^{2x} + \tfrac32$

Integrals from a given derivative

Example 5●●●

Differentiate $xe^x$, and hence find $\displaystyle\int xe^x\,dx$.

$\dfrac{d}{dx}xe^x = xe^x + e^x$.

Differentiate

So $\displaystyle\int (xe^x + e^x)\,dx = xe^x$, which means $\displaystyle\int xe^x\,dx = xe^x - \int e^x\,dx$.

Reverse it

$\displaystyle\int xe^x\,dx = xe^x - e^x + C$

Flashcards

$displaystyleint e^{ax + b},dx$

$\dfrac1a e^{ax + b} + C$

$displaystyleint f'(x),e^{f(x)},dx$

$e^{f(x)} + C$

$e^{ln a}$

$a$

Practice

$\displaystyle\int e^{3x}\,dx = $
$\displaystyle\int e^{-x}\,dx = $
Evaluate $\displaystyle\int_0^{\ln 3} e^x\,dx$.
Evaluate $\displaystyle\int_0^1 e^{2x}\,dx$. Type $e^2$ as e^2.
$\displaystyle\int 2xe^{x^2}\,dx = $
Evaluate $\displaystyle\int_0^1 xe^{x^2}\,dx$. Type $e$ as e.
$f'(x) = e^{2x}$ and $f(0) = 2$. Find $f(\ln 2)$.
Use Example 5 to evaluate $\displaystyle\int_0^1 xe^x\,dx$.
Evaluate $\displaystyle\int_0^{\ln 2} \big(e^x + e^{-x}\big)\,dx$.