Trigonometric Substitution Notes

Published on: September 29, 2026

This Trigonometric Substitution Calculator integrates expressions containing a square root of a sum or difference of squares, and shows the working line by line. Replacing x with a trigonometric function lets a Pythagorean identity collapse the root away, which is the whole reason for the substitution. The answer is converted back to x at the end. Following the same pattern every time keeps the substitution matched to the shape of the root.

Step-by-step method

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

Formula:

\(\begin{gathered} \sqrt{a^{2} - x^{2}}: \; x = a\sin\left(\theta\right) \\ \sqrt{a^{2} + x^{2}}: \; x = a\tan\left(\theta\right) \\ \sqrt{x^{2} - a^{2}}: \; x = a\sec\left(\theta\right) \end{gathered}\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

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

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

Step 2

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

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

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

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

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

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

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

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

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

\(\int \frac{1}{\sqrt{x^{2} + 4}}\, dx = \operatorname{arsinh}{\left(\frac{x}{2} \right)} + C\)
See Example 3 Hide Example 3

Problem

\(\int \sqrt{25 - x^{2}}\, dx\)

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

In this problem: The integrand contains the radical form \(\sqrt{25 - x^{2}}\) with \(a = 5\).

\(\int \sqrt{25 - x^{2}}\, dx\)

Step 2

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

Step 3 - Simplify the radical with the Pythagorean identity.

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

\(\begin{gathered} x = 5\sin\left(\thet5\right),\quad dx = 5\cos\left(\thet5\right)\, d\theta \\ \sqrt{25 - x^{2}} = 5\cos\left(\theta\right) \end{gathered}\)

Step 4

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:

\(\int \sqrt{25 - x^{2}}\, dx = \frac{x \sqrt{25 - x^{2}}}{2} + \frac{25 \arcsin{\left(\frac{x}{5} \right)}}{2} + C\)

Final Answer

\(\int \sqrt{25 - x^{2}}\, dx = \frac{x \sqrt{25 - x^{2}}}{2} + \frac{25 \arcsin{\left(\frac{x}{5} \right)}}{2} + C\)
See Example 4 Hide Example 4

Problem

\(\int \frac{1}{\sqrt{x^{2} + 9}}\, dx\)

Approach

Step-by-step method

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

Step 1

Step 1 - Set up the problem.

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

\(\int \frac{1}{\sqrt{x^{2} + 9}}\, dx\)

Step 2

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

Step 3 - Simplify the radical with the Pythagorean identity.

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

\(\begin{gathered} x = 3\tan\left(\thet3\right),\quad dx = 3\sec^{2}\left(\thet3\right)\, d\theta \\ \sqrt{9 + x^{2}} = 3\sec\left(\theta\right) \end{gathered}\)

Step 4

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:

\(\int \frac{1}{\sqrt{x^{2} + 9}}\, dx = \operatorname{arsinh}{\left(\frac{x}{3} \right)} + C\)

Final Answer

\(\int \frac{1}{\sqrt{x^{2} + 9}}\, dx = \operatorname{arsinh}{\left(\frac{x}{3} \right)} + C\)