Integrating Trigonometric Functions
Integrating trigonometric functions reverses the trig derivatives: \(\displaystyle\int\sin x\,dx=-\cos x+C\) and \(\displaystyle\int\cos x\,dx=\sin x+C\) (radians).
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Integral calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
Integrating trig functions reverses their derivatives (radians). This Year 12 Mathematics Advanced topic (MAV-12-05) 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.