5.2.5 Integration by Substitution

Introduction to Integration by Substitution

Integration by substitution is a powerful technique for solving integrals where a direct approach is challenging. It involves substituting part of the integral with a simpler variable to make the integral easier to solve. For functions involving expressions like $\sqrt{a^2 - x^2}$ or $\sqrt{x^2 - a^2}$ , trigonometric substitutions are particularly useful.

Trigonometric Substitutions for Integrals

For $\sqrt{a^2 - x^2}$ :

Use the substitution:

x = a \sin \theta, \quad dx = a \cos \theta \, d\theta, \quad \sqrt{a^2 - x^2} = a \cos \theta

This substitution is derived from the Pythagorean identity:

\sin^2 \theta + \cos^2 \theta = 1

For $\sqrt{x^2 - a^2}$ :

Use the substitution:

x = a \sec \theta, \quad dx = a \sec \theta \tan \theta \, d\theta, \quad \sqrt{x^2 - a^2} = a \tan \theta

This substitution is based on the identity:

\sec^2 \theta - 1 = \tan^2 \theta

For $\sqrt{x^2 + a^2}$ :

Use the substitution:

x = a \tan \theta, \quad dx = a \sec^2 \theta \, d\theta, \quad \sqrt{x^2 + a^2} = a \sec \theta

This substitution uses:

1 + \tan^2 \theta = \sec^2 \theta

Worked Examples

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Example 1: Evaluate $\int \sqrt{4 - x^2} \, dx$

Step 1: Substitute $x = 2\sin \theta$ :

x = 2\sin \theta, \quad dx = 2\cos \theta \, d\theta, \quad \sqrt{4 - x^2} = 2\cos \theta

Step 2: Rewrite the integral:

\int \sqrt{4 - x^2} \, dx = \int 2\cos \theta \cdot 2\cos \theta \, d\theta = \int 4\cos^2 \theta \, d\theta

Step 3: Simplify using a trigonometric identity:

Use $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$ :

\int 4\cos^2 \theta \, d\theta = \int 2(1 + \cos 2\theta) \, d\theta = 2\int 1 \, d\theta + 2\int \cos 2\theta \, d\theta

Step 4: Integrate term by term:

$\int 1 \, d\theta = \theta$
$\int \cos 2\theta \, d\theta = \frac{\sin 2\theta}{2}$ Combine:

\int 4\cos^2 \theta \, d\theta = 2\theta + \sin 2\theta + C

Step 5: Back-substitute:

Using $\sin \theta = \frac{x}{2}$

and $\cos \theta = \sqrt{1-\sin^2 \theta} = \sqrt{1 - \frac{x^2}{4}} = \frac{\sqrt{4-x^2}}{2}$

$\theta = \arcsin\left(\frac{x}{2}\right)$
$\sin 2\theta = 2\sin \theta \cos \theta = 2\left(\frac{x}{2}\right)\left(\frac{\sqrt{4-x^2}}{2}\right) = \frac{x\sqrt{4-x^2}}{2}.$ Substitute back:

\int \sqrt{4 - x^2} \, dx = 2\arcsin\left(\frac{x}{2}\right) + \frac{x\sqrt{4-x^2}}{2} + C

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Example 2: Evaluate $\int \frac{dx}{\sqrt{x^2 - 9}}$

Step 1: Substitute $x = 3\sec \theta$

x = 3\sec \theta, \quad dx = 3\sec \theta \tan \theta \, d\theta, \quad \sqrt{x^2 - 9} = 3\tan \theta

Step 2: Rewrite the integral:

\int \frac{dx}{\sqrt{x^2 - 9}} = \int \frac{3\sec \theta \tan \theta \, d\theta}{3\tan \theta}

Step 3: Simplify:

\int \frac{dx}{\sqrt{x^2 - 9}} = \int \sec \theta \, d\theta

Step 4: Integrate $sec⁡θ\sec \theta$ :

\int \sec \theta \, d\theta = \ln|\sec \theta + \tan \theta| + C

Step 5: Back-substitute:

Using $x = 3\sec \theta$ , we have:

$\sec \theta = \frac{x}{3}$
$\tan \theta = \sqrt{\sec^2 \theta - 1} = \sqrt{\left(\frac{x}{3}\right)^2 - 1} = \frac{\sqrt{x^2 - 9}}{3}$ Substitute back:

\int \frac{dx}{\sqrt{x^2 - 9}} = \ln\left|\frac{x}{3} + \frac{\sqrt{x^2 - 9}}{3}\right| + C = \ln\left|\frac{x + \sqrt{x^2 - 9}}{3}\right| + C

lightbulbExample

Example 3: Evaluate $\int \frac{dx}{x^2 + 4}$

Step 1: Substitute $x = 2\tan \theta$

x = 2\tan \theta, \quad dx = 2\sec^2 \theta \, d\theta, \quad x^2 + 4 = 4\sec^2 \theta

Step 2: Rewrite the integral:

\int \frac{dx}{x^2 + 4} = \int \frac{2\sec^2 \theta \, d\theta}{4\sec^2 \theta}

Step 3: Simplify:

\int \frac{dx}{x^2 + 4} = \frac{1}{2} \int d\theta

Step 4: Integrate:

\int \frac{dx}{x^2 + 4} = \frac{\theta}{2} + C

Step 5: Back-substitute:

Using $x = 2\tan \theta$ , we have $\theta = \arctan\left(\frac{x}{2}\right)$ :

\int \frac{dx}{x^2 + 4} = \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C

Note Summary

infoNote

Common Mistakes:

Forgetting the substitution derivatives: Ensure $dx$ is replaced correctly.
Incorrect trigonometric identities: Verify $\sin^2 \theta + \cos^2 \theta = 1, \sec^2 \theta - 1 = \tan^2 \theta$ , etc.
Not simplifying fully after back-substitution: Return to the original variable $x$ wherever possible.
Mismanaging bounds in definite integrals: Adjust integration limits to match the substitution.

infoNote

Key Formulas:

Trigonometric Substitutions:

$\sqrt{a^2 - x^2}: x = a\sin \theta, dx = a\cos \theta \, d\theta$
$\sqrt{x^2 - a^2}: x = a\sec \theta, dx = a\sec \theta \tan \theta \, d\theta$
$\sqrt{x^2 + a^2}: x = a\tan \theta, dx = a\sec^2 \theta \, d\theta$

Standard Integrals:

$\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C$
$\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left|x + \sqrt{x^2 - a^2}\right| + C$
$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$

Integration by Substitution Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

5.2.5 Integration by Substitution

Introduction to Integration by Substitution

Trigonometric Substitutions for Integrals

For $\sqrt{a^2 - x^2}$ :

For $\sqrt{x^2 - a^2}$ :

For $\sqrt{x^2 + a^2}$ :

Worked Examples

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Integration by Substitution For their A-Level Exams.

Other Revision Notes related to Integration by Substitution you should explore

Integration by Substitution Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

5.2.5 Integration by Substitution

Introduction to Integration by Substitution

Trigonometric Substitutions for Integrals

For a2−x2\sqrt{a^2 - x^2}a2−x2​:

For x2−a2\sqrt{x^2 - a^2}x2−a2​:

For x2+a2\sqrt{x^2 + a^2}x2+a2​:

Worked Examples

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Integration by Substitution For their A-Level Exams.

Other Revision Notes related to Integration by Substitution you should explore

For $\sqrt{a^2 - x^2}$ :

For $\sqrt{x^2 - a^2}$ :

For $\sqrt{x^2 + a^2}$ :