Welcome to Section 7-3! If you have ever stared at an integral containing a square root like $\sqrt{a^2 - x^2}$ and realized that standard $u$-substitution just won't work, you are in the right place. In this lesson, we explore Trigonometric Substitution, a powerful technique that allows us to trade difficult algebraic radicals for manageable trigonometric identities.
The Core Concept: Getting Rid of the Radical
The main goal of this section is to eliminate the root sign. We do this by taking advantage of the Pythagorean identities (like $\sin^2 \theta + \cos^2 \theta = 1$). By substituting $x$ with a specific trigonometric function, the expression inside the square root becomes a perfect square, allowing us to simplify the integral significantly.
The Three Cases of Substitution
As detailed in the lecture notes, there are three specific patterns to look for. Choosing the correct substitution depends entirely on the form of the radical in your integrand:
- Case 1: $\sqrt{a^2 - x^2}$
When the constant comes first, we use the substitution: $$x = a \sin \theta$$ Why? Because $1 - \sin^2 \theta = \cos^2 \theta$, which clears the square root. - Case 2: $\sqrt{a^2 + x^2}$
When you have a sum of squares, we use the substitution: $$x = a \tan \theta$$ Why? Because $1 + \tan^2 \theta = \sec^2 \theta$. - Case 3: $\sqrt{x^2 - a^2}$
When the variable comes first, we use the substitution: $$x = a \sec \theta$$ Why? Because $\sec^2 \theta - 1 = \tan^2 \theta$.
The Strategy: The Reference Triangle
One of the most important steps in this process—and one highlighted in Example 1 of the notes—is converting your answer back to $x$ after integrating. Since your integral will result in a function of $\theta$ (like $\cot \theta$ or $\csc \theta$), you need a way to translate that back.
Draw a Right Triangle!
- Use your original substitution (e.g., $\sin \theta = \frac{x}{3}$) to label two sides of a right triangle.
- Use the Pythagorean Theorem to find the length of the third side.
- Read the value of the required trig function directly from the triangle (using SOH CAH TOA).
A Note on Definite Integrals
As seen in Example 2 (finding the area of an ellipse) and Example 6 of the class notes, when dealing with definite integrals, you have a choice. You can either change the limits of integration from $x$-values to $\theta$-values (which often saves time), or integrate fully, convert back to $x$ using the triangle method, and use the original limits.
Don't be intimidated by the length of these problems. While they require several steps, the process is systematic. Be sure to review the attached PDF for full step-by-step solutions to the examples.