Do You Have Even More Substitution Examples?

The in-person class has spent some time at the board doing substitution problems. I took pictures so that you would have more examples handy. Several of these examples are similar to ones you all have had trouble with on the homework.

There is a lot of good work here…any mistakes? Which solution is easiest to read…and perhaps the best?

Example 1    $latex \displaystyle \int{\left( x+1 \right){{\left( {{x}^{2}}+2x \right)}^{2}}dx}$


Example 2     $latex \displaystyle \int{4x{{e}^{{{x}^{2}}+9}}dx}$


Example 3     $latex \displaystyle \int{6x{{e}^{2{{x}^{2}}+1}}dx}$


Example 4      $latex \displaystyle \int{5x{{e}^{10{{x}^{2}}+4}}dx}$


Example 5     $latex \displaystyle \int{\frac{\sqrt{2+\ln \left( x \right)}}{x}dx}$


Example 6     $latex \displaystyle \int{\left( {{x}^{3}}+2x \right){{\left( {{x}^{4}}+4{{x}^{2}}+7 \right)}^{8}}dx}$


Example 7      $latex \displaystyle \int{\left( 6-6z \right){{e}^{2z-{{z}^{2}}}}dz}$


Example 8     $latex \displaystyle \int{\left( x+1 \right){{\left( {{x}^{2}}+2x \right)}^{3}}dx}$


Example 9     $latex \displaystyle \int{\left( 3{{x}^{2}}+4 \right){{\left( 2{{x}^{3}}+8x \right)}^{19}}dx}$


How Do You Find an Antiderivative Using the Substitution Method?

In Monday’s class, student found many antiderivatives using the Substitution Method. The basic process is illustrated below for the antiderivative

$latex \displaystyle \int{4{{\left( {{x}^{2}}-3 \right)}^{3}}\cdot 2x dx}$.

Let’s look at the basic steps.

  1. Choose the expression for u. This is generally the inside part of a composition in the integrand. Use the derivative to find an other expression for du.
  2. Match the integrand with u and du. All variables in the original integrand must change to u.
  3. Change the integrand so that it is written in terms of u.
  4. Work out the antiderivative in terms of u.
  5. Put in the expression for u so that the antiderivative is written in terms of the original variable.

Now let’s look at the examples carried out in class.

Problem 1

$latex \displaystyle \int{{{\left( 3{{x}^{2}}+4 \right)}^{4}}\cdot 6x dx}$

Problem 2

$latex \displaystyle \int{{{\left( 2{{x}^{2}}+4 \right)}^{3}}\cdot 4x dx}$

Problem 3

$latex \displaystyle \int{{{\left( {{x}^{2}}-1 \right)}^{5}}\cdot x dx}$

Problem 4

$latex \displaystyle \int{4{{\left( 2x+3 \right)}^{2}} dx}$

Problem 5

$latex \displaystyle \int \frac{2}{{{\left( 2m+1 \right)}^{3}}} dm$

Problem 6

$latex \displaystyle \int{\frac{3}{\sqrt{3u-5}} du}$

Problem 7

$latex \displaystyle \int{-4{{e}^{2p}} dp}$

Problem 8

$latex \displaystyle \int{5{{e}^{-0.3g}} dg}$