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