$Project Greater Ed Logo$

Using U-Sub with Exponential Functions

Business Calculus The U-Substitution Method Using U-Sub with Exponential Functions

Recall How Derivatives of Exponents Work

Suppose $f(x)=e^{x^2+1}$. How does one go about deriving this function?

First, we note that the exponent $x^2+1$ is the “inside function” which is being plugged into $e^x$. So, to derive this we need to use the chain rule. The work and result of this is shown below.

deriving the function with all different parts identified

U-Sub With Exponentials

Suppose now that we are given the task of finding

$$\int 3x^2\cdot e^{x^3}\ dx$$

To find this indefinite integral, we can “undo” the chain rule using u-substitution, noting that the “inside function” that we will let be our $u$ is the exponent $x^3$. The full process for this is shown below.

carrying out the substitution in the integrand

antideriving e^u in terms of u to get e^u

All of the problems you see below essentially work the same way. You might need to do a bit of algebraic manipulation of the $\frac{du}{dx}$ equation so that you can carry out your substitution correctly.

Compute the following antiderivatives

Let $u=x^2$.

First find an “inside function” and let it be called $u$.
Find the derivative of $u$, $\frac{du}{dx}$
Find where the expressions you have for $u$ and $\frac{du}{dx}$ appear in the integral and replace them in the integrand with $u$ and $\frac{du}{dx}$. You may need to first rearrange the equations for $u$ or $\frac{du}{dx}$ before substituting. Your integral should have no $x$’s anymore after subbing.
integrate as usual. Don’t forget “+C”
plug back in your $u$ formula and call it a day
(If computing a definite integral, then plug in upper and lower limits in for $x$, and subtract the results.)

$e^{x^2}+C$

Let $u=2x^3+1$

First find an “inside function” and let it be called $u$.
Find the derivative of $u$, $\frac{du}{dx}$
Find where the expressions you have for $u$ and $\frac{du}{dx}$ appear in the integral and replace them in the integrand with $u$ and $\frac{du}{dx}$. You may need to first rearrange the equations for $u$ or $\frac{du}{dx}$ before substituting. Your integral should have no $x$’s anymore after subbing.
integrate as usual. Don’t forget “+C”
plug back in your $u$ formula and call it a day
(If computing a definite integral, then plug in upper and lower limits in for $x$, and subtract the results.)

$e^{2x^3+1}+C$

Let $u=x^2+x$.

First find an “inside function” and let it be called $u$.
Find the derivative of $u$, $\frac{du}{dx}$
Find where the expressions you have for $u$ and $\frac{du}{dx}$ appear in the integral and replace them in the integrand with $u$ and $\frac{du}{dx}$. You may need to first rearrange the equations for $u$ or $\frac{du}{dx}$ before substituting. Your integral should have no $x$’s anymore after subbing.
integrate as usual. Don’t forget “+C”
plug back in your $u$ formula and call it a day
(If computing a definite integral, then plug in upper and lower limits in for $x$, and subtract the results.)

$e^{x^2+x}+C$

Let $u=(\ln(x)+1)^2$

First find an “inside function” and let it be called $u$.
Find the derivative of $u$, $\frac{du}{dx}$
Find where the expressions you have for $u$ and $\frac{du}{dx}$ appear in the integral and replace them in the integrand with $u$ and $\frac{du}{dx}$. You may need to first rearrange the equations for $u$ or $\frac{du}{dx}$ before substituting. Your integral should have no $x$’s anymore after subbing.
integrate as usual. Don’t forget “+C”
plug back in your $u$ formula and call it a day
(If computing a definite integral, then plug in upper and lower limits in for $x$, and subtract the results.)

$\frac{1}{2}e^{(\ln(x)+1)^2}+C$

Previous Topic

Back to Lesson

Next Topic