Integrals

Integral Forms & Antiderivatives

Basic Forms

1. $\int udv=uv-\int vdu$
   
2. $\int u^{n}du=\frac{u^{n+1}}{n+1}+C,(n\ne -1)$
   
3. $\int \frac{du}{u}=\ln |u|+C$
   
4. $\int e^{u}du=e^u+C$
   
5. $\int a^{u}du=\frac{a^u}{\ln a}+C$
   
6. $\int \sin udu=-\cos u+C$
   
7. $\int \cos udu=\sin u+C$
   
8. $\int {\sec }^{2}udu=\tan u+C$
   
9. $\int {\csc }^{2}udu=-\cot u+C$
   
10. $\int \sec u\tan udu=\sec u+C$
    
11. $\int \csc u\cot udu=-\csc u+C$
    
12. $\int \tan udu=\ln |\sec u|+C$
    
13. $\int \cot udu=\ln |\sin u|+C$
    
14. $\int \sec udu=\ln |\sec u+\tan u|+C$
    
15. $\int \csc udu=\ln |\csc u-\cot u|+C$
    
16. $\int \frac{du}{\sqrt{a^2-u^2}}={\sin }^{-1}\frac{u}{a}+C,(a>0)$
    
17. $\int \frac{du}{a^2+u^2}=\frac{1}{a}{\tan }^{-1}\frac{u}{a}+C$
    
18. $\int \frac{du}{u\sqrt{u^2-a^2}}=\frac{1}{a}{\sec }^{-1}\frac{u}{a}+C$
    
19. $\int \frac{du}{a^2-u^2}=\frac{1}{2a}\ln \left|\frac{u+a}{u-a}\right|+C$
    
20. $\int \frac{du}{u^2-a^2}=\frac{1}{2a}\ln \left|\frac{u-a}{u+a}\right|+C$
    

To Do

• Antiderivative DB search system.
• Find a use for the white-space on the left of the page.