Integrating Trigonometric Functions
Learn integrating trigonometric functions for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-05). Each trig integral reverses a known derivative, so \(\sin x\) integrates to \(-\cos x\), \(\cos x\) to \(\sin x\) and \(\sec^2 x\) to \(\tan x\), always working in radians.
You will learn to integrate sine, cosine and sec squared, including forms like \(\sin(ax + b)\) by dividing by \(a\), handle composite forms with the derivative of the inside present, and evaluate definite integrals with exact values — essential for areas and wave models.
Theory
Integrating trig functions reverses their derivatives (radians). This Year 12 Mathematics Advanced topic integrates \(\sin\), \(\cos\) and \(\sec^{2}\).
\(\displaystyle\int\cos x\,dx=\sin x+C\), \(\displaystyle\int\sin x\,dx=-\cos x+C\), \(\displaystyle\int\sec^{2}x\,dx=\tan x+C\).
With a coefficient, \(\displaystyle\int\cos(ax)dx=\dfrac1a\sin(ax)+C\), and similarly for \(\sin\). Angles are in radians.
Method
- Recall the base integral.
- Divide by the coefficient of \(x\) if present.
- Work in radians; add \(+C\) or evaluate limits.
| \(\ \) | \(=\) | \(\sin x+C\) |
| \(\ \) | \(=\) | \(-\dfrac12\cos 2x+C\) |
| \(\ \) | \(=\) | \(\tan x+C\) |
| \(\ \) | \(=\) | \(\big[\sin x\big]_0^{\pi/2}=1\) |
Common pitfalls
Frequently asked questions
What is the integral of cos x?
It is sin x plus C.
What is the integral of sin x?
It is minus cos x plus C.
How do you integrate sin 2x?
It is minus one half cos 2x plus C, because you divide by the coefficient 2.
What is the area under y = sin x from 0 to pi?
It is 2, found from minus cos evaluated between 0 and pi.