Trigonometric Integrals Calculator

This Trigonometric Integrals Calculator helps you solve trigonometric integrals and shows each step clearly. It can be used to integrate expressions involving powers and products of sine, cosine, and other trigonometric functions by applying standard identities, substitutions, and integration techniques where needed. This makes it useful for checking answers, understanding how trigonometric integration works, and practising calculus step by step.

Step-by-step method

1. Write the integral.
2. Identify the trig form and powers.
3. Pick the correct trig-integral strategy.
4. Rewrite using identities (if needed).
5. Use substitution where appropriate.
6. Integrate and simplify.
7. Add + C.

Formula bank:

Example 1: sin(x)^3*cos(x)^2, x

Step 1 - Write the integral.

In this problem: We will apply trig-integral patterns.

Step 3 - Identify the powers.

In this problem: sin power m = 3, cos power n = 2.

Step 4 - Pick the strategy.

In this problem: Odd power of sin(x): save one sin(x), convert the rest using sin²(x)=1−cos²(x).

Step 5 - Rewrite the integrand.

In this problem: Save one sin(x) for du and rewrite remaining sin powers.

Step 6 - Substitute.

In this problem: Let u = cos(x), so du = −sin(x)dx.

Step 7 - Integrate in u.

In this problem: Integrate the polynomial in u.

Step 8 - Back-substitute and finish.

In this problem: Replace u with cos(x) and add + C.

Final answer: cos(x)^5/5 - cos(x)^3/3 + C

Example 2: tan(x)^5*sec(x)^2, x

Step 1 - Write the integral.

In this problem: We will apply trig-integral patterns.

Step 3 - Identify the powers.

In this problem: tan power m = 5, sec power n = 2.

Step 4 - Pick the strategy.

In this problem: Even power of sec(x): save sec²(x)dx and convert remaining sec powers using sec²=1+tan².

Step 5 - Rewrite the integrand.

In this problem: Save sec²(x)dx for du.

Step 6 - Substitute.

In this problem: Let u = tan(x), so du = sec²(x)dx.

Step 7 - Integrate in u.

In this problem: Integrate the polynomial in u.

Step 8 - Back-substitute and finish.

In this problem: Replace u with tan(x) and add + C.

Final answer: tan(x)^6/6 + C

Example 3: sin(x)^4, x

Step 1 - Write the integral.

In this problem: We will apply trig-integral patterns.

Step 3 - Identify the powers.

In this problem: sin power m = 4, cos power n = 0.

Step 4 - Pick the strategy.

In this problem: Both powers are even: use power-reduction identities, rewrite to a sum, then integrate.

Step 5 - Rewrite to a sum.

In this problem: Convert even powers into a trig sum (multiple angles).

Step 6 - Integrate and finish.

In this problem: Integrate term-by-term and add + C.

Final answer: 3x/8 - sin(2x)/4 + sin(4x)/32 + C