Table of Contents

1. Introduction
2. Integration by Substitution
   1. Basic Steps for $u$-Substitution
   2. Worked Example 1
3. Advanced $u$-Substitution
   1. Worked Example 2
4. Definite Integrals Using $u$-Substitution
   1. Worked Example 3
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

Introduction

In calculus, integration by substitution (also known as $u$-substitution) is a method used to find integrals. It simplifies an integral by transforming it into an easier one through a change of variable.

Integration by Substitution

Basic Steps for $u$-Substitution

What is integration by substitution?

Substitution simplifies an integral by defining an alternative variable (usually $u$) in terms of the original variable (usually $x$). The new integral in $u$ is often easier to solve, and the substitution can be reversed at the end to express the answer in terms of $x$.

How do I integrate simple functions using $u$-substitution?

In a simple integral involving substitution, you will usually be integrating a composite function (i.e., 'function of a function'). Substitution can be a safer method when integrating by inspection is awkward or difficult to spot.

Steps for $u$-Substitution:

1. Identify the substitution to be used – it will be the secondary (or 'inside') function in a composite function.

   • If the integral involves $f(g(x))$, let $u = g(x)$.
   • Example: $\int 3x \cos(x^2 - 5) , dx$
     • Let $u = x^2 - 5$.

2. Differentiate the substitution and rearrange.

   • $\frac{du}{dx}$ can be treated like a fraction. Multiply by $dx$ to get rid of fractions.
   • Example: $u = x^2 - 5 \Rightarrow \frac{du}{dx} = 2x$
     • Then $du = 2x , dx \Rightarrow x , dx = \frac{1}{2} , du$.

3. Replace all parts of the integral.

   • Replace all $u$ terms, including $dx$.
   • Example: $\int 3x \cos(x^2 - 5) , dx = 3 \int \cos(x^2 - 5) \cdot x , dx$
     • So $\int 3x \cos(x^2 - 5) , dx = 3 \int \cos(u) \cdot \frac{1}{2} , du = \frac{3}{2} \int \cos(u) , du$.

4. Integrate.

   • Example: $\frac{3}{2} \int \cos(u) , du = \frac{3}{2} \sin(u) + C$
     • Don't forget the constant of integration.

5. Substitute $x$ back in.

   • Replace $u$ everywhere with the equivalent expression for $x$.
   • Example: $\frac{3}{2} \sin(u) + C = \frac{3}{2} \sin(x^2 - 5) + C$
     • So $\int 3x \cos(x^2 - 5) , dx = \frac{3}{2} \sin(x^2 - 5) + C$.

Worked Example 1

Find the indefinite integral $\int \frac{x^2 - 1}{2x^3 - 6x + 7} , dx$.

Solution:

1. Identify the substitution:

   • Let $u = 2x^3 - 6x + 7$.

2. Differentiate and rearrange:

   • $\frac{du}{dx} = 6x^2 - 6 \Rightarrow du = 6(x^2 - 1) , dx \Rightarrow (x^2 - 1) , dx = \frac{1}{6} , du$.

3. Replace all parts of the integral:

   • $\int \frac{x^2 - 1}{2x^3 - 6x + 7} , dx = \int \frac{1}{u} \cdot \frac{1}{6} , du = \frac{1}{6} \int \frac{1}{u} , du$.

4. Integrate:

   • $\frac{1}{6} \int \frac{1}{u} , du = \frac{1}{6} \ln|u| + C$.

5. Substitute $x$ back in:

   • $\int \frac{x^2 - 1}{2x^3 - 6x + 7} , dx = \frac{1}{6} \ln|2x^3 - 6x + 7| + C$.

Advanced $u$-Substitution

The procedure here is exactly the same as for integrating simpler functions, but the substitution to use may not be as obvious.

How do I integrate more complicated functions using $u$-substitution?

• Practice more complex questions to improve your integration by substitution skills.
  • Example: $\int x \sqrt{x - 4} , dx$
    • Identify the substitution: Let $u = x - 4$.
    • Differentiate and rearrange: $\frac{du}{dx} = 1 \Rightarrow du = dx$.
    • Replace all parts of the integral: $u = x - 4 \Rightarrow x = u + 4$.
      • So $\int x \sqrt{x - 4} , dx = \int (u + 4) \sqrt{u} , du$.
    • Integrate: $\int (u + 4) \sqrt{u} , du = \int (u + 4) u^{\frac{1}{2}} , du = \int (u^{\frac{3}{2}} + 4u^{\frac{1}{2}}) , du = \frac{2}{5} u^{\frac{5}{2}} + \frac{8}{3} u^{\frac{3}{2}} + C$.
    • Substitute $x$ back in: $\int x \sqrt{x - 4} , dx = \frac{2}{5} (x - 4)^{\frac{5}{2}} + \frac{8}{3} (x - 4)^{\frac{3}{2}} + C$.

Worked Example 2

Find the indefinite integral $\int \frac{dx}{\sqrt{16 - 9x^2}}$.

Solution:

1. Identify the substitution:

   • Recall the standard integral $\int \frac{dx}{\sqrt{1 - x^2}} = \arcsin(x) + C$.
   • Rearrange the integral: $\int \frac{dx}{\sqrt{16 - 9x^2}} = \int \frac{dx}{4\sqrt{1 - \frac{9}{16}x^2}} = \frac{1}{4} \int \frac{dx}{\sqrt{1 - \frac{9}{16}x^2}}$.

2. Substitute:

   • Let $u = \frac{3}{4}x \Rightarrow u^2 = \frac{9}{16}x^2$.

3. Differentiate and rearrange:

   • $\frac{du}{dx} = \frac{3}{4} \Rightarrow dx = \frac{4}{3} , du$.

4. Replace all parts of the integral:

   • $\frac{1}{4} \int \frac{dx}{\sqrt{1 - \frac{9}{16}x^2}} = \frac{1}{4} \int \frac{\frac{4}{3} , du}{\sqrt{1 - u^2}} = \frac{1}{3} \int \frac{du}{\sqrt{1 - u^2}}$.

5. Integrate:

   • $\frac{1}{3} \int \frac{du}{\sqrt{1 - u^2}} = \frac{1}{3} \arcsin(u) + C$.

6. Substitute $x$ back in:

   • $\int \frac{dx}{\sqrt{16 - 9x^2}} = \frac{1}{3} \arcsin\left(\frac{3}{4}x\right) + C$.

Definite Integrals Using $u$-Substitution

How do I evaluate definite integrals using $u$-substitution?

Definite integrals can also be solved using $u$-substitution. You just need to rewrite the integration limits in terms of $u$ as well.

Example: $\int_4^8 x \sqrt{x - 4} , dx$

1. Substitution:

   • Let $u = x - 4$.

2. Differentiate and rearrange:

   • $\frac{du}{dx} = 1 \Rightarrow du = dx$.

3. Change the integral limits:

   • When $x = 4$, $u = 4 - 4 = 0$.
   • When $x = 8$, $u = 8 - 4 = 4$.

4. Replace all parts of the integral:

   • $\int_4^8 x \sqrt{x - 4} , dx = \int_0^4 (u + 4) \sqrt{u} , du$.

5. Integrate and evaluate the definite integral using the $u$ values:

   • $\int_0^4 (u + 4) \sqrt{u} , du = \left[ \frac{2}{5} u^{\frac{5}{2}} + \frac{8}{3} u^{\frac{3}{2}} \right]_0^4 = \frac{2}{5} \cdot 32 + \frac{8}{3} \cdot 8 - \left( \frac{2}{5} \cdot 0 + \frac{8}{3} \cdot 0 \right) = \frac{512}{15}$.

Worked Example 3

Evaluate the definite integral $\int_1^2 \frac{2x + 3}{3x^2 + 9x - 5} , dx$.

Solution:

1. Identify the substitution:

   • Let $u = 3x^2 + 9x - 5$.

2. Differentiate and rearrange:

   • $\frac{du}{dx} = 6x + 9 \Rightarrow du = 3(2x + 3) , dx \Rightarrow (2x + 3) , dx = \frac{1}{3} , du$.

3. Find the integration limits in terms of $u$:

   • When $x = 1$, $u = 3(1)^2 + 9(1) - 5 = 7$.
   • When $x = 2$, $u = 3(2)^2 + 9(2) - 5 = 25$.

4. Replace all parts of the integral, including the integration limits:

   • $\int_1^2 \frac{2x + 3}{3x^2 + 9x - 5} , dx = \int_7^{25} \frac{1}{u} \cdot \frac{1}{3} , du = \frac{1}{3} \int_7^{25} \frac{1}{u} , du$.

5. Integrate and evaluate the definite integral:

   • $\frac{1}{3} \int_7^{25} \frac{1}{u} , du = \frac{1}{3} \left[ \ln u \right]_7^{25} = \frac{1}{3} (\ln 25 - \ln 7)$.

6. Use laws of logarithms to simplify the final answer:

   • $\int_1^2 \frac{2x + 3}{3x^2 + 9x - 5} , dx = \frac{1}{3} \ln\left(\frac{25}{7}\right)$.

Practice Questions

Practice Question

1. Find the indefinite integral $\int x e^{x^2} , dx$.

2. Evaluate the definite integral $\int_0^1 \frac{2x}{(x^2 + 1)^2} , dx$.

3. Find the indefinite integral $\int \frac{\sin(\ln x)}{x} , dx$.

4. Evaluate the definite integral $\int_1^4 \frac{3x + 2}{(x^2 + 2x + 1)} , dx$.

5. Find the indefinite integral $\int \frac{x^3}{x^2 + 1} , dx$.

Glossary

• Composite Function: A function composed of two functions such that the output of one function becomes the input of the other.
• Constant of Integration: An arbitrary constant added to the function after integration, denoted by $C$.
• Definite Integral: An integral with upper and lower limits, which gives the area under the curve between those limits.
• Indefinite Integral: An integral without limits, representing a family of functions.
• Substitution: A method used to simplify an integral by introducing a new variable.

Summary and Key Takeaways

Key Points:

• Substitution is a powerful method for simplifying integrals, especially for composite functions.
• Identify the inner function in a composite function to make the correct substitution.
• Differentiate and rearrange the substitution to replace all parts of the integral.
• Integrate the simplified integral and always remember to substitute back the original variable.

Key Takeaways:

• Practice identifying appropriate substitutions in various integrals.
• Remember to change the limits of integration when evaluating definite integrals.
• Regular practice with $u$-substitution will improve your problem-solving skills in calculus.

Exam Tips:

Common Mistakes: