CAS header

 

Find this content useful?
Think it should stay ad-free?

Integration of Trigonometric Functions

Here we look at the integration of basic trig functions. These results can be extended by application to integration by substitution and integration by parts.

Integration of Trigonometric Functions

As an immediate consequence of the results on differentiation, we have $$ \int \sin(x)dx = -\cos(x) + c $$ $$ \int \cos(x)dx = \sin(x) + c $$ Application of a substitution yields $$ \int \sin(kx)dx = -\frac{1}{k}\cos(x) + c $$ $$ \int \cos(kx)dx = \frac{1}{k}\sin(kx) + c $$

Integration of Trigonometric Functions 2

Evaluate the following integral $$\int \sin(3t)\cos(3t)dt $$
solution - press button to display
The key trick in this integration is the application of the double angle formula. $$\begin{align} \int \sin(3t)\cos(3t)dt&= \int \frac{1}{2}\sin(6t)dt \\ &= -\frac{1}{12}\cos(6t)+ c \end{align} $$

Integration of Trigonometric Functions 3

Evaluate the integral $$ \int \sin t \cos^3t dt $$
solution - press button to display
Let $u = \cos t$ then $du = -\sin t dt$. Substitution of these terms yields $$ \int \sin (t) \cos^3(t) dt = \int -u^3du = -\frac{1}{4}u^{4} + c $$