Trig Sub Integral Calculator

Trigonometric substitution is a powerful technique in calculus for evaluating integrals that contain square roots of quadratic expressions. This method transforms the integrand into a form that can be integrated using standard trigonometric identities. Our calculator provides a step-by-step solution for integrals that require trigonometric substitution.

What is Trigonometric Substitution?

Trigonometric substitution is a method used to evaluate integrals that contain square roots of quadratic expressions. The technique involves substituting a trigonometric function for the variable in the integrand, which simplifies the expression and makes it integrable.

The most common forms of trigonometric substitution are:

Substitution for √(a² - x²): Use x = a sinθ
Substitution for √(x² - a²): Use x = a secθ
Substitution for √(x² + a²): Use x = a tanθ

Each substitution is chosen based on the form of the integrand. After substitution, the integral can be evaluated using standard trigonometric identities and techniques.

How to Use This Calculator

Our trigonometric substitution integral calculator provides a step-by-step solution for integrals that require this technique. To use the calculator:

Enter the integral expression in the input field. For example, ∫(1/√(9 - x²)) dx
Select the appropriate substitution type from the dropdown menu
Click "Calculate" to see the step-by-step solution
Review the result and the detailed steps

The calculator will show the substitution used, the transformed integral, and the final result. You can also view a graphical representation of the integral if available.

Key Formulas

The following formulas are commonly used in trigonometric substitution:

For √(a² - x²): x = a sinθ dx = a cosθ dθ √(a² - x²) = a cosθ

For √(x² - a²): x = a secθ dx = a secθ tanθ dθ √(x² - a²) = a tanθ

For √(x² + a²): x = a tanθ dx = a sec²θ dθ √(x² + a²) = a secθ

These substitutions transform the original integral into a form that can be evaluated using standard trigonometric identities and techniques.

Worked Examples

Example 1: ∫(1/√(9 - x²)) dx

This integral uses the substitution x = 3 sinθ. The steps are:

Let x = 3 sinθ, dx = 3 cosθ dθ
When x = 0, θ = 0; when x = 3, θ = π/2
Substitute into the integral: ∫(1/√(9 - 9 sin²θ)) * 3 cosθ dθ = ∫(1/3 cosθ) * 3 cosθ dθ = ∫1 dθ
Integrate: θ + C
Back-substitute θ = arcsin(x/3): arcsin(x/3) + C

The final result is arcsin(x/3) + C.

Example 2: ∫(1/√(x² - 4)) dx

This integral uses the substitution x = 2 secθ. The steps are:

Let x = 2 secθ, dx = 2 secθ tanθ dθ
When x = 2, θ = 0; when x = ∞, θ = π/2
Substitute into the integral: ∫(1/√(4 sec²θ - 4)) * 2 secθ tanθ dθ = ∫(1/2 tanθ) * 2 secθ tanθ dθ = ∫secθ dθ
Integrate: ln|secθ + tanθ| + C
Back-substitute θ = arccos(x/2): ln|x/2 + √(x²/4 - 1)| + C

The final result is ln|x/2 + √(x²/4 - 1)| + C.

FAQ

When should I use trigonometric substitution?

Use trigonometric substitution when your integral contains a square root of a quadratic expression, especially when the expression can be rewritten in the form √(a² - x²), √(x² - a²), or √(x² + a²).

What substitution should I use for √(a² - x²)?

For integrals containing √(a² - x²), use the substitution x = a sinθ. This transforms the integrand into a form that can be integrated using standard trigonometric identities.

How do I know which substitution to use?

The substitution you use depends on the form of the integrand. If the integrand contains √(a² - x²), use x = a sinθ. If it contains √(x² - a²), use x = a secθ. If it contains √(x² + a²), use x = a tanθ.