### Math Notes

Subjects

#### Integral Calculus Solutions

##### Topics || Problems

Integration is also called as anti-derivatives

I. Basic Integration Formulas

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

II. Inverse Trigonometric Functions

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

III. Hyperbolic / Inverse Hyperbolic Functions

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