Trigonometric Substitution Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Plus

This Trigonometric Substitution Calculator integrates expressions containing radicals such as √(a^2 − x^2), √(a^2 + x^2), or √(x^2 − a^2). It picks the matching sine, tangent, or secant substitution, simplifies the radical with a Pythagorean identity, integrates, and converts back to x.

Step-by-step method

  1. Set up the integral and find the radical form.
  2. Choose the matching trigonometric substitution.
  3. Simplify the radical with the Pythagorean identity.
  4. Integrate and convert back to x, adding C.

Formula:

\(\begin{gathered} \sqrt{a^{2} - x^{2}}: \; x = a\sin\left(\theta\right) \\ \sqrt{a^{2} + x^{2}}: \; x = a\tan\left(\theta\right) \\ \sqrt{x^{2} - a^{2}}: \; x = a\sec\left(\theta\right) \end{gathered}\)

Example 1:

\(\int \sqrt{9 - x^{2}}\, dx\)

Step 1 - Set up the integral and find the radical form.

In this problem: The integrand contains the radical form \(\sqrt{9 - x^{2}}\) with \(a = 3\).

\(\int \sqrt{9 - x^{2}}\, dx\)

Step 2 - Choose the matching trigonometric substitution.

In this problem: Match the radical to one of the three standard forms and substitute.

\(\begin{gathered} \sqrt{a^{2} - x^{2}}: \; x = a\sin\left(\theta\right) \\ \sqrt{a^{2} + x^{2}}: \; x = a\tan\left(\theta\right) \\ \sqrt{x^{2} - a^{2}}: \; x = a\sec\left(\theta\right) \end{gathered}\)

Step 3 - Simplify the radical with the Pythagorean identity.

In this problem: With this substitution the radical collapses through the Pythagorean identity.

\(\begin{gathered} x = 3\sin\left(\thet3\right),\quad dx = 3\cos\left(\thet3\right)\, d\theta \\ \sqrt{9 - x^{2}} = 3\cos\left(\theta\right) \end{gathered}\)

Step 4 - Integrate and convert back to x, adding C.

In this problem: After integrating in θ and converting back to x, the antiderivative is:

\(\int \sqrt{9 - x^{2}}\, dx = \frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \arcsin{\left(\frac{x}{3} \right)}}{2} + C\)

Final answer:

\(\int \sqrt{9 - x^{2}}\, dx = \frac{x \sqrt{9 - x^{2}}}{2} + \frac{9 \arcsin{\left(\frac{x}{3} \right)}}{2} + C\)

Example 2:

\(\int \frac{1}{\sqrt{x^{2} + 4}}\, dx\)

Step 1 - Set up the integral and find the radical form.

In this problem: The integrand contains the radical form \(\sqrt{x^{2} + 4}\) with \(a = 2\).

\(\int \frac{1}{\sqrt{x^{2} + 4}}\, dx\)

Step 2 - Choose the matching trigonometric substitution.

In this problem: Match the radical to one of the three standard forms and substitute.

\(\begin{gathered} \sqrt{a^{2} - x^{2}}: \; x = a\sin\left(\theta\right) \\ \sqrt{a^{2} + x^{2}}: \; x = a\tan\left(\theta\right) \\ \sqrt{x^{2} - a^{2}}: \; x = a\sec\left(\theta\right) \end{gathered}\)

Step 3 - Simplify the radical with the Pythagorean identity.

In this problem: With this substitution the radical collapses through the Pythagorean identity.

\(\begin{gathered} x = 2\tan\left(\thet2\right),\quad dx = 2\sec^{2}\left(\thet2\right)\, d\theta \\ \sqrt{4 + x^{2}} = 2\sec\left(\theta\right) \end{gathered}\)

Step 4 - Integrate and convert back to x, adding C.

In this problem: After integrating in θ and converting back to x, the antiderivative is:

\(\int \frac{1}{\sqrt{x^{2} + 4}}\, dx = \operatorname{arsinh}{\left(\frac{x}{2} \right)} + C\)

Final answer:

\(\int \frac{1}{\sqrt{x^{2} + 4}}\, dx = \operatorname{arsinh}{\left(\frac{x}{2} \right)} + C\)