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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

Problem 1 – Introduction

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

Use the table below to guide you.

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

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

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

4. sin(x)cos(x)dx\begin{align*}\sin (x) \cos (x) dx\end{align*} can be rewritten as 12 sin(2x)\begin{align*}\frac{1}{2} \ \sin(2x)\end{align*} using the Double Angle formula.

What is the result when you integrate 12 sin(2x)\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. x+1x2+2x+3dx\begin{align*}\int\limits \frac{x+1}{x^2+2x+3} dx\end{align*}

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

7. x4x2+1dx\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. tan(x)dx\begin{align*}\int\limits \tan(x) dx\end{align*}

10. cos3(x)\begin{align*}\int\limits \cos^3 (x)\end{align*}

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