Paul's Online Notes
Paul's Online Notes
Home / Calculus I / Integrals / Substitution Rule for Indefinite Integrals
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 5-3 : Substitution Rule for Indefinite Integrals

1. Evaluate \( \displaystyle \int{{\left( {8x - 12} \right){{\left( {4{x^2} - 12x} \right)}^4}\,dx}}\).

Show All Steps Hide All Steps

Hint : Recall that if there is a term in the integrand (or a portion of a term) with an “obvious” inside function then there is at least a chance that the “inside” function is the substitution that we need.
Start Solution

In this case it looks like we should use the following as our substitution.

\[u = 4{x^2} - 12x\]
Hint : Recall that after the substitution all the original variables in the integral should be replaced with \(u\)’s.
Show Step 2

Because we need to make sure that all the \(x\)’s are replaced with \(u\)’s we need to compute the differential so we can eliminate the \(dx\) as well as the remaining \(x\)’s in the integrand.

\[du = \left( {8x - 12} \right)\,dx\]

Recall that, in most cases, we will also need to do a little manipulation of the differential prior to doing the substitution. In this case we don’t need to do that.

Show Step 3

Doing the substitution and evaluating the integral gives,

\[\int{{\left( {8x - 12} \right){{\left( {4{x^2} - 12x} \right)}^4}\,dx}} = \int{{{u^4}\,du}} = \frac{1}{5}{u^5} + c\]
Hint : Don’t forget that the original variable in the integrand was not \(u\)!
Show Step 4

Finally, don’t forget to go back to the original variable!

\[\int{{\left( {8x - 12} \right){{\left( {4{x^2} - 12x} \right)}^4}\,dx}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{5}{{\left( {4{x^2} - 12x} \right)}^5} + c}}\]