# Basic Integral Formulas

1) $\int {1dx = x + c}$

2) $\int {adx = ax + c}$ Where $a$is any constant.

3) $\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + c}$

4) $\int {{{[f(x)]}^n}f'(x)dx = \frac{{{{[f(x)]}^{n + 1}}}}{{n + 1}}} + c$

5) $\int {\frac{1}{x}dx = \ln x + c}$

6) $\int {\frac{{f'(x)}}{{f(x)}}dx = \ln f(x) + c}$

7) $\int {{a^x}dx = \frac{{{a^x}}}{{\ln x}} + c}$

8) ${\int a ^{f(x)}}dx = \frac{{{a^{f(x)}}}}{{\ln a}} + c$

9) $\int {{e^x}dx = {e^x} + c}$

10) $\int {{e^{f(x)}}dx = {e^{f(x)}} + c}$

11) $\int {af(x)dx = a\int {f(x)} }$

12) $\int {[f(x) \pm g(x)]dx = \int {f(x)dx \pm \int {g(x)dx} } }$

13) $\int {f(x) \cdot g(x)dx = f(x)\left( {\int {g(x)dx} } \right) – \left[ {f'(x)\left( {\int {g(x)dx} } \right)} \right]dx}$

14) $\int {\ln xdx = x(\ln x – 1) + c}$

15) $\int {\sin xdx = – \cos x + c}$

16) $\int {\cos xdx = \sin x + c}$

17) $\int {\tan xdx = \ln \sec x} + c$ or $– \ln \cos x + c$

18) $\int {\cot xdx = \ln \sin x + c}$

19) $\int {\sec xdx = \ln (\sec x + \tan x) + c}$ or $\ln \tan \left( {\frac{x}{2} + \frac{\pi }{4}} \right) + c$

20) $\int {\csc xdx = \ln (\csc x – \cot x) + c}$ or $\ln \tan \frac{x}{2} + c$

21) $\int {{{\sec }^2}xdx = \tan x + c}$

22) $\int {{{\csc }^2}xdx = – \cot x + c}$

23) $\int {\sec x\tan xdx = \sec x + c}$

24) $\int {\csc x\cot xdx = – \csc x + c}$

25) $\int {\sinh xdx = \cosh x + c}$

26) $\int {\cosh xdx = \sinh x + c}$

27) $\int {\tanh xdx = \ln \cosh x + c}$

28) $\int {\coth xdx = \ln \sinh x + c}$

29) $\int {\sec {\text{h}}xdx = {{\tan }^{ – 1}}(\sinh x) + c}$

30) $\int {\csc {\text{h}}xdx = – {{\coth }^{ – 1}}(\cosh x)}$

31) $\int {\sec {{\text{h}}^2}xdx = \tanh x + c}$

32) $\int {\csc {{\text{h}}^2}xdx = – \coth x + c}$

33) $\int {\sec {\text{h}}x\tanh xdx = – \sec {\text{h}}x + c}$

34) $\int {\csc {\text{h}}x\coth xdx = – \csc {\text{h}}x + c}$

35) $\int {\frac{1}{{\sqrt {{a^2} – {x^2}} }}dx = {{\sin }^{ – 1}}\frac{x}{a}} + c$ or ${\cos ^{ – 1}}\frac{x}{a} + c$

36) $\int {\frac{1}{{\sqrt {{x^2} – {a^2}} }}dx = {{\cosh }^{ – 1}}\frac{x}{a}} + c$ or $\ln (x + \sqrt {{x^2} – {a^2}} ) + c$

37) $\int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = {{\sinh }^{ – 1}}\frac{x}{a} + c}$ or $\ln (x + \sqrt {{x^2} + {a^2}} ) + c$

38) $\int {\frac{1}{{{a^2} – {x^2}}}dx = \frac{1}{a}{{\tanh }^{ – 1}}\frac{x}{a} + c}$ or $\frac{1}{{2a}}\ln \left( {\frac{{a + x}}{{a – x}}} \right) + c$

39) $\int {\frac{1}{{{x^2} – {a^2}}}dx = – \frac{1}{a}{{\coth }^{ – 1}}\frac{x}{a} + c}$ or $\frac{1}{{2a}}\ln \left( {\frac{{x – a}}{{x + a}}} \right) + c$

40) $\int {\frac{1}{{{x^2} + {a^2}}}dx = \frac{1}{a}{{\tan }^{ – 1}}\frac{x}{a} + c}$