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Indefinite Integral Formulas

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Rules

1. Multiplication by constant

   $\int kf\left(x\right) dx=k\int f\left(x\right) dx$

2. Addition/Subtraction

   $\int \left[f\right(x)\pm g(x\left)\right] dx=\int f\left(x\right) dx\pm \int g\left(x\right) dx$

3. Integration by parts

   $\int f(x)g^{\prime }(x) dx=f(x)g(x)-\int f^{\prime }(x)g(x) dx$

4. Integration by substitution

   $i. u=u(x),\; so, du=u^{\prime }(x) dx$
   $ii. x=x(u),\; so, dx=x^{\prime }(u) du$

   Using i and/or ii, solve the integral and then substitute back

   $\int f(x) dx=\int f(x(u)) x^{\prime }(u) du$
   or $\int f(x) dx=\int f(x(u)) \frac{1}{u^{\prime }(x)}du$

Basic Indefinite Integrals

5. $\int k dx=kx+c$
6. $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c,n\ne -1$

Trignometric Indefinite Integrals

7. $\int \sin (ax) dx=-\frac{\cos \left(ax\right)}{a}+c$
8. $\int \cos (ax) dx=\frac{\sin \left(ax\right)}{a}+c$
9. $\int {\sin }^2(ax) dx=\frac{x}{2}-\frac{\sin \left(2ax\right)}{4a}+c$
10. $\int {\cos }^2(ax) dx=\frac{x}{2}+\frac{\sin \left(2ax\right)}{4a}+c$
11. $\int {\sec }^{2}x dx=\tan x+c$
12. $\int {\csc }^{2}x dx=-\cot x+c$
13. $\int \sec x \tan x dx=\sec x+c$
14. $\int \csc x \cot x dx=-\csc x+c$
15. $\int x \sin (ax) dx=\frac{\sin \left(ax\right)-ax \cos \left(ax\right)}{a^2}+c$
16. $\int x \cos (ax) dx=\frac{\cos \left(ax\right)+ax \sin \left(ax\right)}{a^2}+c$
17. $\int \tan x dx=-\ln |\cos x|+c$
18. $\int \cot x dx=\ln |\sin x|+c$
19. $\int \sec x dx=\ln |\sec x+\tan x|+c$
20. $\int \csc x dx=-\ln |\csc x+\cot x|+c$

Exponential Indefinite Integrals

21. $\int e^{ax}dx=\frac{e^{ax}}{a}+c$
22. $\int a^{x}dx=\frac{a^x}{\ln \left(a\right)}+c,a>0 \& a\ne 1$
23. $\int \ln x dx=x \ln \left(x\right)-x+c$
24. $\int x^m{e}^{ax}dx=\frac{x^m{e}^{ax}}{a}-\frac{m}{a}\int x^{m-1}{e}^{ax}dx$
25. $\int x{e}^{ax}dx=\frac{x{e}^{ax}}{a}-\frac{e^{ax}}{a^2}=\frac{(ax-1) e^{ax}}{a^2}$

Fractional Indefinite Integrals

26. $\int \frac{1}{x} dx=\ln \left|x\right|+c$
27. $\int \frac{1}{\sqrt{a^2-x^2}} dx={\sin }^{-1}\left(\frac{x}{a}\right)+c$
28. $\int \frac{1}{a^2+x^2} dx=\frac{{\tan }^{-1}\left(\frac{x}{a}\right)}{a}+c$
29. $\int \frac{1}{x \sqrt{x^2-a^2}} dx=\frac{{\sec }^{-1}\left(\frac{x}{a}\right)}{a}+c$

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