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Integrating Trig Functions Online Calculator

Trig integral calculator. Enter a trig expression and interval to integrate numerically, supporting power-reduction and double-angle practice.

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∫ sin²x dx = x/2 − sin2x/4
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Integrating Trig Functions

1. Definition
Trig integrals use basic formulas + identities. Even powers use power reduction (sin²x=(1−cos2x)/2); products use product-to-sum; odd powers keep one factor for substitution.
2. Common formulas
SymbolsMeaning
∫sin x−cos x+C
∫cos xsin x+C
∫sin²xx/2 − sin2x/4+C
∫cos²xx/2 + sin2x/4+C
∫tan x−ln|cos x|+C
3. How it works
  • Adaptive Simpson integrates the trig expression directly;
  • It verifies power-reduction / product-to-sum results;
  • Avoid poles of tan and cot.
4. Steps
  1. Enter a trig expression, e.g. sin(x)^2;
  2. Enter [a,b];
  3. Click Calculate for the numeric integral.
5. Example
Example: ∫₀^π sin²x dx = π/2 ≈ 1.5708. Tool: 1.570796.
6. Pitfalls
sin(x)^2 means sin²x, not sin(x²);
tan and cot have poles at π/2, 0, etc.;
Reduce even powers first.

❓ FAQ | Integrating Trig Functions

sin(x)^2 vs sin(x^2)?
The former is (sin x)², the latter is sin(x²); they differ.
How to integrate even powers of sin?
Use sin²x=(1−cos2x)/2 repeatedly, then integrate term by term.
Does it simplify analytically?
No; it only gives numbers to compare with.
Why does tan fail?
tan is undefined at π/2+kπ; avoid those poles.
Which functions?
sin cos tan sec cot and +−*/^ etc.