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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 $\int\limits \sqrt{2x+3}dx$. Let $u = 2x + 3$. Evaluate the integral using substitution.

Use the table below to guide you.

$f(x) =$ $\sqrt{2x+3}$
$u =$ $2x+3$
$du =$
$g (u) =$
$\int\limits g(u)du =$
$\int\limits f(x)dx =$

2. Try using substitution to integrate $\int\limits \sin (x) \cos (x) dx$. Let $u = \sin(x)$.

3. Now integrate the same integral, but let $u = \cos(x)$. How does the result compare to the one above?

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

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

## Problem 2 – Common Feature

Find the result of the following integrals using substitution.

5. $\int\limits \frac{x+1}{x^2+2x+3} dx$

6. $\int\limits \sin(x) \ e^{\cos(x)} dx$

7. $\int\limits \frac{x}{4x^2+1}dx$

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. $\int\limits \tan(x) dx$

10. $\int\limits \cos^3 (x)$

Feb 23, 2012

Nov 04, 2014