Calculus — Integrals⏱ 7 min read

Trigonometric Integrals

∫ sin x dx = −cos x + C

What is the Trigonometric Integrals?

These ten integrals are each the reverse of the corresponding trigonometric derivative rule. The integrals of tan, cot, sec, and csc all involve logarithms — a consequence of the substitution technique used to derive them, which reduces each to the form ∫(1/u)du = ln|u|.

tan²x integrates to tan x − x + C, derived using the Pythagorean identity tan²x = sec²x − 1 to rewrite it as a difference of two simpler integrals.

What Each Variable Means

∫ sin x dx

Integral of sineEquals −cos x + C.

∫ tan x dx

Integral of tangentEquals ln|sec x| + C — involves a logarithm, unlike sin and cos.

∫ sec²x dx

Integral of sec squaredEquals tan x + C — the direct reverse of tan x's own derivative.

When to Use It

Finding the antiderivative of any single trigonometric function
Evaluating definite integrals over an interval involving trig functions
As building blocks for more complex trigonometric integration problems

Step-by-Step Examples

Example 1: A definite integral

Problem: Evaluate ∫₀^(π/2) cos x dx

Find the antiderivative

The antiderivative of cos x is sin x.

[sin x]₀^(π/2)

Apply the limits

Evaluate at both bounds and subtract.

sin(π/2) − sin(0) = 1 − 0

✓

Answer: 1

Example 2: Deriving ∫tan²x dx

Problem: Show that ∫ tan²x dx = tan x − x + C, using the identity tan²x = sec²x − 1

Replace tan²x using the identity

Rewrite the integrand.

∫ tan²x dx = ∫ (sec²x − 1) dx

Integrate each term separately

Split into two simpler integrals.

= ∫ sec²x dx − ∫ 1 dx

✓

Answer: = tan x − x + C

Common Mistakes

Mistake: Assuming ∫sin x dx = cos x + C (missing the negative sign).
Fix: ∫sin x dx = −cos x + C. Checking by differentiating −cos x gives back sin x, confirming the negative sign is required.
Mistake: Forgetting the logarithms for tan, cot, sec, and csc.
Fix: Unlike sin and cos, these four integrate to logarithmic expressions (e.g. ∫tan x dx = ln|sec x| + C), not simple trig functions.

Practice Questions

Evaluate ∫ sec²x dx.
Evaluate ∫₀^π sin x dx.
Hint: The antiderivative is -cos x; -cos(π) - (-cos(0)) = 1 - (-1).

Frequently Asked Questions

How are these integrals connected to the trig derivatives?

Each is the exact reverse: since d/dx(sin x) = cos x, reversing gives ∫cos x dx = sin x + C. Since d/dx(tan x) = sec²x, reversing gives ∫sec²x dx = tan x + C.

Why do tan, cot, sec, and csc integrate to logarithms?

Each can be rewritten and solved via u-substitution in a form that reduces to ∫(1/u)du, which always integrates to ln|u| + C.