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 — knowing the six basic trig derivatives cold makes most of this table predictable rather than something to memorize separately. 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|, the same 1/x-family result that shows up in basic integration.

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 — a pattern worth remembering, since squared trig functions almost always need a Pythagorean identity rewrite before they can be integrated directly.

These integrals show up constantly in physics and engineering contexts involving periodic motion (pendulums, waves, alternating current), not just in a pure math course — anywhere a sine or cosine wave needs to be summed or averaged over an interval, one of these ten results is usually doing the work underneath.

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.