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Integration Techniques 1-Substitution

Today, I am going to begin to look at integrals using the different integration techniques shown in Calculus 2. For the next week, I will be giving a new post everyday describing how to evaluate a different integral. By the end of the week we will have an example of each of the integration techniques.

Hopefully you will join me each day as I go through these integrals in both a post and in a video. Along the way, I hope you learn something, and, if you are taking Calculus 2, you are in a great position to do very well on any upcoming exams you may have. You can find more Calculus 2 study help posts here. I also have a decision tree available so that you can follow along as we determine what technique of integration to use. I also have a useful list of formula sheet available.

 

Find \(\int \cos^{2}(x)\sin(x)dx\).

In order to find this integral, we start by looking at our decision tree.

  • Can you guess the answer.
    Here I will work with the assumption that we cannot guess what the answer is going to be.
  • Is there an inside function.
    Yes, there is indeed an inside function. We can pick \(i=\cos(x)\). In order to continue with substitution, we then find that \(di=-\sin(x)dx\) and \(dx=\frac{-di}{\sin(x)}\). This gives us that
    \begin{align*}
    \int \cos^{2}(x)\sin(x)dx&=\int i^{2}\sin(x)\frac{-di}{\sin(x)} \\
    &=-\int i^{2} di.
    \end{align*}
  • Since we were able to cancel all of the \(x\)s, we begin the decision tree again for the new integral.
  • For \(\int i^{2}di\), we should be able to guess the answer with the power rule, and we find that
    \begin{align*}
    \int i^{2}di =\frac{i^{3}}{3}+c.
    \end{align*}
  • Next, we make sure this is correct by finding the derivative of our answer. We see that \(\frac{d}{di}(\frac{i^{3}}{3}+c)=\frac{3i^{2}}{3}=i^{2}\). Therefore, we have integrated correctly.
  • In order to solve our initial problem, we now use this and substitute \(i=\cos(x)\) to get the answer,
    \begin{align*}
    \int \cos^{2}(x)\sin(x)dx&=-\frac{i^{3}}{3}+c \\
    &=-\frac{\cos^{3}(x)}{3}+c.
    \end{align*}

Therefore, we have found the integral we were looking for.

Conclusion

Here we were able to find a solution using substitution.  While it may also have been possible to complete this integral using another technique, I have set up the decision tree to always try substitution first because it is almost always the easiest techniques to use.  Because we have an answer, there is no reason to try another technique, so we would move onto our next problem.

I hope you learned something from this post and found it helpful.  If you did, make sure to share it and the video with anyone else you know that may need some extra help with integrals.

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