Integrals of Trigonometric Functions
| Function |
Integral |
| sinx |
-cosx + c |
| cosx |
sinx + c |
| sin2x |
x/2 - sin(2x)/4 + c = (x - sinx ∙ cosx)/2 + c |
| cos2x |
x/2 + sin(2x)/4 + c = (x + sinx ∙ cosx)/2 + c |
| tanx = sec2x |
-ln|cosx| + c |
| cotx = -csc2x |
ln|sinx| + c |
| secx |
ln|secx + tanx| + c |
| cscx |
-ln|cscx + cotx| + c |
| sec2x |
tanx + c |
| csc2x |
-cotx + c |
Example 1: Calculate the following integral ∫x2 sinx3dx.
Solution:
∫x2 sinx3dx = ∫ sinx3 x2 dx
Set u = x3 and du = 3x2dx or du/3 = x2dx, then we have:
∫x2 sinx3 dx
= ∫sinu du/3
= 1/3 * ∫sinu du
= 1/3 *(-cosu) + C
= 1/3 *(-cosx3) + C
Example 2: Calculate 
Solution:
Let u = ln t. So du = (1/t) dt.
We then have:
Example 3: Evaluate ∫(3sin x 4sec2x) dx
Solution:
∫(3sinx 4sec2x) dx
= 3∫sinxdx - 4∫sec2x dx
= -3cosx – 4tanx + C
Example 4: Integrate ∫(2+ tanx)2 dx
Solution:
