Rules of Integration of Trigonometric Functions

In what follows $c$ is a constant of integration.

Rules of Integration Involving Trigonometric Functions

1. $\int \sin (x)\,dx = -\cos (x) + c$
   
   
2. $\int \cos (x)\,dx = \sin (x) + c$
   
   
3. $\int \tan (x)\,dx = -\ln| \cos (x)| + c$
   
   
4. $\int \cot (x)\,dx = \ln| \sin (x)| + c$
   
   
5. $\int \sec (x)\,dx = \ln| \sec (x) + tan (x)| + c$
   
   
6. $\int \csc (x)\,dx = -\ln| \csc (x) + cot (x)| + c$
   
   also
   
   
7. $\int \sec^2 (x)\,dx = \tan (x) + c$
   because $\dfrac{d}{dx}(\tan(x))= sec^2 (x)$
   
   
8. $\int \csc^2 (x)\,dx = -\cot (x) + c$
   because $\dfrac{d}{dx}(\cot(x))= -csc^2 (x)$
   
   
9. $\int \sec(x) \tan(x)\,dx = \sec (x) + c$
   because $\dfrac{d}{dx}(\sec(x))= sec(x) \tan (x)$
   
   
10. $\int \csc(x) \cot(x)\,dx = -\csc (x) + c$
    because $\dfrac{d}{dx}(\csc(x))= -\csc(x)\cot(x)$