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8.1: Integration by Substitution

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Calculus, Chapter 7, Lesson 1.

In this activity, you will explore:

  • Integration of standard forms
  • Substitution methods of integration

Use this document to record your answers. Check your answers with the Integrate command.

Problem 1 – Introduction

1. Consider the integral \begin{align*}\int\limits \sqrt{2x+3}dx\end{align*}2x+3dx. Let \begin{align*}u = 2x + 3\end{align*}. Evaluate the integral using substitution.

Use the table below to guide you.

\begin{align*}f(x) =\end{align*} \begin{align*}\sqrt{2x+3}\end{align*}
\begin{align*}u = \end{align*} \begin{align*}2x+3\end{align*}
\begin{align*}du =\end{align*}
\begin{align*}g (u) =\end{align*}
\begin{align*}\int\limits g(u)du = \end{align*}
\begin{align*}\int\limits f(x)dx = \end{align*}

2. Try using substitution to integrate \begin{align*}\int\limits \sin (x) \cos (x) dx\end{align*}. Let \begin{align*}u = \sin(x)\end{align*}.

3. Now integrate the same integral, but let \begin{align*}u = \cos(x)\end{align*}. How does the result compare to the one above?

4. \begin{align*}\sin (x) \cos (x) dx\end{align*} can be rewritten as \begin{align*}\frac{1}{2} \ \sin(2x)\end{align*} using the Double Angle formula.

What is the result when you integrate \begin{align*}\int\limits \frac{1}{2} \ \sin(2x)\end{align*} using substitution?

Problem 2 – Common Feature

Find the result of the following integrals using substitution.

5. \begin{align*}\int\limits \frac{x+1}{x^2+2x+3} dx\end{align*}

6. \begin{align*}\int\limits \sin(x) \ e^{\cos(x)} dx\end{align*}

7. \begin{align*}\int\limits \frac{x}{4x^2+1}dx\end{align*}

8. What do these integrals have in common that makes them suitable for the substitution method?

Extension

Use trigonometric identities to rearrange the following integrals and then use the substitution method to integrate.

9. \begin{align*}\int\limits \tan(x) dx\end{align*}

10. \begin{align*}\int\limits \cos^3 (x)\end{align*}

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Feb 23, 2012
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TI.MAT.ENG.SE.1.Calculus.8.1

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