# Integrals of Exponential Functions Calculator

## Get detailed solutions to your math problems with our Integrals of Exponential Functions step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

### Difficult Problems

1

Solved example of integrals of exponential functions

$\int\left(2x+7\right)e^{x^2+7x}dx$
2

We can solve the integral $\int e^{\left(x^2+7x\right)}\left(2x+7\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x^2+7x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=x^2+7x$

Find the derivative

$\frac{d}{dx}\left(x^2+7x\right)$

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(7x\right)$

The derivative of the linear function times a constant, is equal to the constant

$\frac{d}{dx}\left(x^2\right)+7$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x+7$
3

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=\left(2x+7\right)dx$
4

Isolate $dx$ in the previous equation

$\frac{du}{\left(2x+7\right)}=dx$

Simplify the fraction $\frac{e^u\left(2x+7\right)}{2x+7}$ by $2x+7$

$\int e^udu$
5

Substituting $u$ and $dx$ in the integral and simplify

$\int e^udu$
6

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$e^u$

$e^{\left(x^2+7x\right)}$
7

Replace $u$ with the value that we assigned to it in the beginning: $x^2+7x$

$e^{\left(x^2+7x\right)}$
8

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$e^{\left(x^2+7x\right)}+C_0$

$e^{\left(x^2+7x\right)}+C_0$