Trigonometric Substitution – Intro Part 1

This lesson kicks off your journey into one of the most creative and conceptually rich techniques in Calculus 2: Trigonometric Substitution.

We begin by building your intuition for when and why trig substitution is the best tool — especially when radicals are involved. You'll learn to identify classic expression forms like:

• √(a² - x²)

• √(x² + a²)

• √(x² - a²)

And you'll master the substitutions that match each one, such as x = a sin(θ), x = a tan(θ), or x = a sec(θ).

We also review key identities you'll rely on throughout this topic:

• Pythagorean Identities

• Derivatives of trig functions

• Half-angle formulas

• How to complete the square (when needed)

By the end of this lecture, you’ll understand the framework and rationale behind trig substitution — and be ready to apply it confidently.

Download all Trig Substitution slides in the Resources tab.

Trigonometric Substitution – Intro Part 2

In this second introduction to Trigonometric Substitution, we turn abstract concepts into concrete strategy. You’ll learn the full substitution workflow, from setup through simplification and back-substitution — all driven by the Pythagorean identities that power this method.

We focus on the three classic radical forms and their substitutions:

• √(a² - x²) → x = a sin(θ) using 1 - sin²(θ) = cos²(θ)

• √(a² + x²) → x = a tan(θ) using 1 + tan²(θ) = sec²(θ)

• √(x² - a²) → x = a sec(θ) using sec²(θ) - 1 = tan²(θ)

You’ll learn:
How to choose the correct substitution based on the structure of the radical
How to rewrite the entire integral in terms of θ, including dx
The role of Pythagorean identities in eliminating the square roots
How and when to draw a right triangle for accurate back-substitution
Why trigonometric substitution simplifies integration in cases where algebra fails

We also walk through a complete worked example using x = a sin(θ), showing every step with clarity.

By the end of this lecture, you’ll understand not only how trig substitution works — but why it’s effective, and exactly what to look for when choosing this technique.

All Trig Substitution slides are available in the Resources tab.

Trigonometric Substitution – Example 1 ( √(x² − a²) with Rational Expression)

In this first worked example, we tackle a challenging integral involving a rational expression and a radical of the form:
√(x² − a²)

We solve:
∫ dx / (x² · √(x² − 4))

You’ll learn:
How to identify this as a √(x² − a²) form, requiring the substitution x = 2 sec(θ)
How to correctly transform both x² and the radical using sec(θ)
How the identity sec²(θ) − 1 = tan²(θ) simplifies the expression
How to substitute and simplify dx
How to complete the integration and back-substitute using a right triangle

This example combines algebra, trig identities, and substitution skills to handle a more complex integrand — the kind that often appears on exams.

All Trig Substitution slides are available in the Resources tab of this section.

Master this one and you're well on your way to dominating trig substitution!

Trigonometric Substitution – Example 2 ( Definite Integral with √(x² − a²))

This example builds on your understanding of √(x² − a²) forms — but adds a key twist: definite integration.

We solve:
∫ from 2 to 3 of dx / [x² · √(x² − 1)]

You’ll learn:
How to identify the proper substitution: x = sec(θ)
How to change the limits of integration from x-values to θ-values
How to apply the identity sec²(θ) − 1 = tan²(θ) to simplify the radical
How to transform the entire integrand and compute the definite integral in θ
Why definite integrals often don’t require back-substitution

This example is essential practice for handling definite integrals with trig substitution — a format frequently tested in both university courses and AP Calculus BC.

All Trig Substitution slides are available in the Resources tab of this section.

You'll finish this example with a stronger grasp of how to navigate limits, substitutions, and identities with confidence.

Trigonometric Substitution – Example 3 ( √(a² − u²) with Coefficients)

In this third example, we revisit the classic √(a² − x²) form — but this time, the integrand includes a coefficient inside the square root, requiring an extra layer of algebraic care.

We solve:
∫ dx / √(4 − 9x²)

You’ll learn:
How to factor the expression under the square root
Why the substitution x = (2/3) sin(θ) works — and how to find it
How to properly adjust dx and simplify the radical using 1 − sin²(θ) = cos²(θ)
How to perform the integration and back-substitute with a triangle

This example highlights an important skill: recognizing when to factor out a constant before applying your trig substitution. You'll see how small algebraic differences can lead to significant changes in your substitution setup.

All Trig Substitution slides are available in the Resources tab of this section.

Trigonometric Substitution – Example 4 (Advanced Challenge with Definite Bounds)

Ready for a challenge? This final example pushes your trig substitution skills to the limit — combining algebraic complexity, trigonometric identities, and definite bounds.

We solve:
∫ from 0 to (3√3)/2 of (x³ / (4x² + 9)^(3/2)) dx

You’ll learn:
How to spot the underlying √(a² + u²) structure and make the substitution x = (3/2) tan(θ)
How to manage powers in both the numerator and radical
How to simplify using 1 + tan²(θ) = sec²(θ) and reduce the expression cleanly
How to change limits from x-values to θ-values and finish the definite integral
How all your previous trig sub skills come together in one integrated problem

This problem is not for the faint of heart — but if you can master this one, you can do any trig substitution. It’s the kind of integral that sets you apart.

All Trig Substitution slides are available in the Resources tab of this section.

If you’ve made it this far, congratulations — you’re building elite-level intuition.

Introduction to Trigonometric Integration Part One

Learn how to set up different types of PFD and the common integrals you will need. Recopy these problems and learn to recognize the setup and practice the step by step solutions for each type.