Let q and c be constants, c be the constant of integration.

1. \(\int {qdx} = q\int {dx} = qx + c\)

2. \(\int {\left[ {f\left( x \right) \pm g\left( x \right)} \right]dx} = \int {f\left( x \right)dx \pm \int {g\left( x \right)} } dx\)

3. \(\int {{x^n}dx = \frac{{{u^{n + 1}}}}{{n + 1}}} \) Power Formula

4. \(\int {{e^u}du = {e^u} + c} \)

5. \(\int {\frac{{du}}{u} = \ln u + c} \)

6. \(\int {\sin udu = - \cos u + c} \)

7. \(\int {\cos udu = \sin u + c} \)

8. \(\int {\tan udu = \ln \left[ {\sec u} \right] + c} \)

9. \(\int {\cot udu = - \ln \left[ {\csc u} \right] + c} \)

10. \(\int {\sec udu = \ln \left[ {\sec u + \tan u} \right]} + c\)

11. \(\int {\csc udu = - \ln \left[ {\csc u + \cot u} \right] + c} \)

12. \(\int {\arcsin udu = u\arcsin u + \sqrt {1 - {u^2}} + C} \)

13. \(\int {\arctan udu = u\arctan u + \ln \sqrt {1 + {u^2}} + C} \)

14. \(\int {\frac{{du}}{{\sqrt {{a^2} - {u^2}} }}} = \arcsin \frac{u}{a} + C\)

15. \(\int {\frac{{du}}{{{a^2} + {u^2}}}} = \frac{1}{a}\arctan \frac{u}{a} + C\)

16. \(\int {\frac{{du}}{{u\sqrt {{u^2} - {a^2}} }}} = \frac{1}{a}arc\sec \frac{u}{a} + C\)

17. \(\int {\frac{{du}}{{\sqrt {{u^2} + {a^2}} }}} = {\mathop{\rm arcsinh}\nolimits} \frac{u}{a} + C\)

18. \(\int {\frac{{du}}{{\sqrt {{u^2} - {a^2}} }}} = {\mathop{\rm arccosh}\nolimits} \frac{u}{a} + C:u > a > 0\)

19. \(\int {\frac{{du}}{{{a^2} - {u^2}}}} = \frac{1}{a}{\mathop{\rm arctanh}\nolimits} \frac{u}{a} + C:{u^2} < {a^2}\)

20. \(\int {\frac{{du}}{{{a^2} - {u^2}}}} = \frac{{ - 1}}{a}{\mathop{\rm arccoth}\nolimits} \frac{u}{a} + C:{u^2} > {a^2}\)

21. \(\int {\sinh udu = \cosh u + C} \)

22. \(\int {\cosh u} du = \sinh u + C\)

23. \(\int {\tanh u} du = \ln \cosh u + C\)

24. \(\int {\coth u} du = \ln \left| {\sinh u} \right| + C\)

25. \(\int {{{\sec }^2}u} du = \tanh u + C\)

26. \(\int {{{\csc }^2}u} du = - \coth u + C\)

27. \(\int {{\mathop{\rm sech}\nolimits} u\tanh u} du = - {\mathop{\rm sech}\nolimits} u + C\)

28. \(\int {{\mathop{\rm csch}\nolimits} u\coth u} du = - {\mathop{\rm csch}\nolimits} u + C\)