Use Trigonometric Substitution to Evaluate The Integral Calculator

What is Trigonometric Substitution?

Trigonometric substitution is a method used to evaluate integrals that contain square roots of quadratic expressions. The key idea is to replace the quadratic expression with a trigonometric function, typically sine or cosine, which can then be integrated using standard trigonometric identities.

The most common forms of trigonometric substitution involve expressions of the form √(a² - x²), √(x² - a²), and √(x² + a²). Each of these forms has a corresponding trigonometric substitution that simplifies the integral.

When to Use Trigonometric Substitution

You should consider using trigonometric substitution when:

• The integrand contains a square root of a quadratic expression.
• Other integration techniques like substitution or integration by parts do not simplify the integral.
• The integral involves trigonometric functions or inverse trigonometric functions.

Trigonometric substitution is most effective when the integrand can be rewritten in terms of a trigonometric function, which allows you to use trigonometric identities to simplify the integral.

Step-by-Step Guide

Step 1: Identify the Type of Integral

First, identify the type of integral you are dealing with. The most common forms are:

• √(a² - x²)
• √(x² - a²)
• √(x² + a²)

Step 2: Choose the Appropriate Substitution

Based on the type of integral, choose the appropriate trigonometric substitution:

Step 3: Perform the Substitution

Substitute the chosen trigonometric function into the integral and adjust the differential dx accordingly.

Step 4: Simplify the Integral

Use trigonometric identities to simplify the integral. This often involves using the Pythagorean identity sin²θ + cos²θ = 1 or sec²θ - 1 = tan²θ.

Step 5: Integrate

Integrate the simplified expression using standard integration techniques.

Step 6: Back-Substitute

After integrating, back-substitute the trigonometric function to express the result in terms of the original variable x.

Common Integrals Solved with Trigonometric Substitution

Here are some common integrals that can be solved using trigonometric substitution:

These formulas are derived using trigonometric substitution and are useful for evaluating integrals involving square roots of quadratic expressions.

Example Calculation

Let's evaluate the integral ∫√(9 - x²) dx using trigonometric substitution.

Step 1: Identify the Type of Integral

The integral involves √(9 - x²), which matches the form √(a² - x²) where a = 3.

Step 2: Choose the Appropriate Substitution

We use x = 3 sinθ.

Step 3: Perform the Substitution

Let x = 3 sinθ, then dx = 3 cosθ dθ.

Step 4: Simplify the Integral

The integral becomes ∫√(9 - 9 sin²θ) * 3 cosθ dθ = ∫3√(1 - sin²θ) * 3 cosθ dθ = ∫9 cos²θ dθ.

Step 5: Integrate

Using the identity cos²θ = (1 + cos2θ)/2, the integral becomes ∫9(1 + cos2θ)/2 dθ = (9/2)∫(1 + cos2θ) dθ = (9/2)(θ + (sin2θ)/2) + C.

Step 6: Back-Substitute

Recall that x = 3 sinθ, so θ = arcsin(x/3). The derivative of arcsin(x/3) is 1/√(9 - x²).

The final result is (9/2)arcsin(x/3) + (9/4)x/√(9 - x²) + C.