Integral Calculus

1. $\displaystyle \int du = u + C$

2. $\displaystyle \int a \, du = a\int du$

3. $\displaystyle \int (du + dv + ... + dz) = \int du + \int dv + ... + \int dz$

4. $\displaystyle \int f (x)\,dx = F(x) + C$

5. $\displaystyle \int_a^b f(x) \, dx = F(b) - F(a)$

6. $\displaystyle \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$

7. $\displaystyle \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$

8. $\displaystyle \int_a^b f(x) \, dx = \int_a^b f(z) \, dz$

9. $\displaystyle \int u^n \, du = \dfrac{u^{n + 1}}{n + 1} + C; \, n \neq -1$

10. $\displaystyle \int \dfrac{du}{u} = \ln u + C$

11. $\displaystyle \int a^u \, du = \dfrac{a^u}{\ln a} + C, \,\, a > 0, \,\, a \neq 1$

12. $\displaystyle \int e^u \, du = e^u + C$

13. $\displaystyle \int \sin u \, du = -\cos u + C$

14. $\displaystyle \int \cos u \, du = \sin u + C$

15. $\displaystyle \int \sec^2 u \, du = \tan u + C$

16. $\displaystyle \int \csc^2 u \, du = -\cot u + C$

17. $\displaystyle \int \sec u \, \tan u \, du = \sec u + C$

18. $\displaystyle \int \csc u \, \cot u \, du = -\csc u + C$

19. $\displaystyle \int \tan u \, du = \ln (\sec u) + C = -\ln (\cos u) + C$

20. $\displaystyle \int \cot u \, du = \ln (\sin u) + C$

21. $\displaystyle \int \sec u \, du = \ln (\sec u + \tan u) + C$

22. $\displaystyle \int \csc u \, du = \ln (\csc u - \cot u) + C = -\ln (\csc u + \cot u) + C$

23. $\displaystyle \int \dfrac{du}{\sqrt{a^2 - u^2}} = \arcsin \, \dfrac{u}{a} + C, \,\,\, a > 0$

24. $\displaystyle \int \dfrac{du}{a^2 + u^2} = \dfrac{1}{a}\arctan \, \dfrac{u}{a} + C$

25. $\displaystyle \int \dfrac{du}{u\sqrt{u^2 - a^2}} = \dfrac{1}{a} {\rm arcsec} \, \dfrac{u}{a} + C$

26. $\displaystyle \int u\,dv = uv - \int v\, du$