# 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 \begin{align*}\int\limits \sqrt{2x+3}dx\end{align*}. 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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