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Calculus⏱ 7 min read

Trigonometric Integrals

∫ sin x dx = −cos x + C

📋 Complete Table of Trig Integrals

All 10 standard trigonometric integrals — each is the reverse of the corresponding derivative rule.

# Integral Result
1 ∫ sin x dx −cos x + C
2 ∫ cos x dx sin x + C
3 ∫ tan x dx ln|sec x| + C
4 ∫ cot x dx ln|sin x| + C
5 ∫ sec x dx ln|sec x + tan x| + C
6 ∫ csc x dx ln|csc x − cot x| + C
7 ∫ sec²x dx tan x + C
8 ∫ sec x tan x dx sec x + C
9 ∫ csc²x dx −cot x + C
10 ∫ tan²x dx tan x − x + C

📝 Why ∫ tan²x dx = tan x − x + C

This is derived using the Pythagorean identity tan²x = sec²x − 1:

1
Replace tan²x using the identity∫ tan²x dx = ∫ (sec²x − 1) dx
2
Integrate each term= ∫ sec²x dx − ∫ 1 dx
✓
= tan x − x + C

📝 Example — Definite Integral

Evaluate: ∫₀^(π/2) cos x dx

1
Antiderivative of cos x is sin x[sin x]₀^(π/2)
2
Apply the limitssin(π/2) − sin(0) = 1 − 0
✓
Answer: 1

💡 Key Pattern — Reverse the Derivatives

Since d/dx(sin x) = cos x, reversing gives ∫ cos x dx = sin x + C
Since d/dx(−cos x) = sin x, reversing gives ∫ sin x dx = −cos x + C
Since d/dx(tan x) = sec²x, reversing gives ∫ sec²x dx = tan x + C
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🔗 Related Formulas

🎯 Quick Fact

The integrals of tan, cot, sec, and csc all involve logarithms. This is because they can be computed via substitution, reducing them to ∫(1/u)du = ln|u|.