Trigonometric Integrals Calculator

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

This Trigonometric Integrals Calculator integrates sine, cosine, tangent and their relatives, and shows the working line by line. Each has its own standard result, and the minus signs land on the opposite functions to where they sit when differentiating. Recognising the form is most of the work. Following the same pattern every time keeps the signs from being carried over from the derivative rules.

Step-by-step method

  1. Set up the problem.
  2. Write the integration rule for that form.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Formula:

\(\begin{gathered} \int \sin\left(x\right)\, dx = -\cos\left(x\right) + C \\ \int \cos\left(x\right)\, dx = \sin\left(x\right) + C \\ \int \sec^{2}\left(x\right)\, dx = \tan\left(x\right) + C \end{gathered}\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration rule for that form.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

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

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

\(\int \sin{\left(x \right)}\, dx = -\cos\left(x\right) + C\)
See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the integration rule for that form.
  3. Apply the rule, keeping the constant multiple in front, and add C.

Step 1

Step 1 - Set up the problem.

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

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

Step 2

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

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

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

Example Problems

Click one to load it into the box and solve it.