Integrals

1. ∫af(x)dx=a∫f(x)dx${\displaystyle \int af\left(x\right)dx}=a{\displaystyle \int f\left(x\right)dx}$
2. ∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx${\displaystyle \int \left[f\left(x\right)+g\left(x\right)\right]dx}={\displaystyle \int f\left(x\right)dx}+{\displaystyle \int g\left(x\right)dx}$
3. ∫xmdx=xm+1m+1(m≠−1)=lnx(m=−1)$\begin{array}{cc}\hfill {\displaystyle \int x^{m}dx}& =\frac{x^{m+1}}{m+1}\left(m\ne \text{−}1\right)\hfill \\ & =\text{ln}x(m=\mathrm{-1})\hfill \end{array}$
4. ∫sinxdx=−cosx${\displaystyle \int \text{sin}xdx}=\text{−}\text{cos}x$
5. ∫cosxdx=sinx${\displaystyle \int \text{cos}xdx}=\text{sin}x$
6. ∫tanxdx=ln|secx|${\displaystyle \int \text{tan}xdx}=\text{ln}\left|\text{sec}x\right|$
7. ∫sin2axdx=x2−sin2ax4a${\displaystyle \int {\text{sin}}^{2}axdx}=\frac{x}{2}-\frac{\text{sin}2ax}{4a}$
8. ∫cos2axdx=x2+sin2ax4a${\displaystyle \int {\text{cos}}^{2}axdx}=\frac{x}{2}+\frac{\text{sin}2ax}{4a}$
9. ∫sinaxcosaxdx=−cos2ax4a${\displaystyle \int \text{sin}ax\text{cos}axdx}=-\frac{\text{cos}2ax}{4a}$
10. ∫eaxdx=1aeax${\displaystyle \int e^{ax}dx}=\frac{1}{a}{e}^{ax}$
11. ∫xeaxdx=eaxa2(ax−1)${\displaystyle \int x{e}^{ax}dx}=\frac{e^{ax}}{a^2}\left(ax-1\right)$
12. ∫lnaxdx=xlnax−x${\displaystyle \int \text{ln}axdx}=x\text{ln}ax-x$
13. ∫dxa2+x2=1atan−1xa${\displaystyle \int \frac{dx}{a^2+x^2}}=\frac{1}{a}{\text{tan}}^{\mathrm{-1}}\frac{x}{a}$
14. ∫dxa2−x2=12aln∣∣x+ax−a∣∣${\displaystyle \int \frac{dx}{a^2-x^2}}=\frac{1}{2a}\text{ln}\left|\frac{x+a}{x-a}\right|$
15. ∫dxa2+x2−−−−−−√=sinh−1xa${\displaystyle \int \frac{dx}{\sqrt{a^2+x^2}}}={\text{sinh}}^{\mathrm{-1}}\frac{x}{a}$
16. ∫dxa2−x2−−−−−−√=sin−1xa${\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}}={\text{sin}}^{\mathrm{-1}}\frac{x}{a}$
17. ∫a2+x2−−−−−−√dx=x2a2+x2−−−−−−√+a22sinh−1xa${\displaystyle \int \sqrt{a^2+x^2}dx}=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}{\text{sinh}}^{\mathrm{-1}}\frac{x}{a}$
18. ∫a2−x2−−−−−−√dx=x2a2−x2−−−−−−√+a22sin−1xa${\displaystyle \int \sqrt{a^2-x^2}dx}=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\text{sin}}^{\mathrm{-1}}\frac{x}{a}$