Trigonometric Integrals Calculator

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

This Trigonometric Integrals Calculator integrates standard trigonometric forms such as sin(x), cos(x), sec(x)^2, and sec(x)tan(x). Each form has a known antiderivative, and any constant multiple stays in front. The constant of integration C is included in the answer.

Step-by-step method

  1. Identify the trigonometric form and any constant multiple.
  2. Write the integration rule for that form.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Formula:

\(\begin{gathered} \int \sin\left(x\right)\, dx = -\cos\left(x\right) + C \\ \int \cos\left(x\right)\, dx = \sin\left(x\right) + C \\ \int \sec^{2}\left(x\right)\, dx = \tan\left(x\right) + C \end{gathered}\)

Example 1:

\(\int \sin{\left(x \right)}\, dx\)

Step 1 - Identify the trigonometric form and any constant multiple.

In this problem: The trigonometric form is \(\sin{\left(x \right)}\).

\(\int \sin{\left(x \right)}\, dx\)

Step 2 - Write the integration rule for that form.

In this problem: Each standard trigonometric form has a known antiderivative.

\(\int \sin{\left(x \right)}\, dx = -\cos\left(x\right) + C\)

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the trigonometric form with its antiderivative and simplify.

\(\int \sin{\left(x \right)}\, dx = -\cos\left(x\right) + C\)

Final answer:

\(\int \sin{\left(x \right)}\, dx = -\cos\left(x\right) + C\)

Example 2:

\(\int \sec^{2}{\left(x \right)}\, dx\)

Step 1 - Identify the trigonometric form and any constant multiple.

In this problem: The trigonometric form is \(\sec^{2}{\left(x \right)}\).

\(\int \sec^{2}{\left(x \right)}\, dx\)

Step 2 - Write the integration rule for that form.

In this problem: Each standard trigonometric form has a known antiderivative.

\(\int \sec^{2}{\left(x \right)}\, dx = \tan\left(x\right) + C\)

Step 3 - Apply the rule, keeping the constant multiple in front, and add C.

In this problem: Replace the trigonometric form with its antiderivative and simplify.

\(\int \sec^{2}{\left(x \right)}\, dx = \tan\left(x\right) + C\)

Final answer:

\(\int \sec^{2}{\left(x \right)}\, dx = \tan\left(x\right) + C\)