Table of integrals

| $\displaystyle \int{{{x}^{n}}dx=\frac{{{x}^{n+1}}}{a+1}+c}$ | $\displaystyle \int{{{u}^{/}}{{u}^{a}}dx=\frac{{{u}^{a+1}}}{a+1}+c}$ | | $\displaystyle \int{\frac{1}{\sqrt{x}}dx=2\sqrt{x}+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{\sqrt{u}}dx=2\sqrt{u}+c}$ | | $\displaystyle \int{{{a}^{x}}}dx=\frac{{{a}^{x}}}{\ln a}+c$ | $\displaystyle \int{{{u}^{/}}{{a}^{u}}}dx=\frac{{{a}^{u}}}{\ln a}+c$ | | $\displaystyle \int{{{e}^{x}}}dx={{e}^{x}}+c$ | | | $\displaystyle \int{\frac{1}{x}}dx=\ln \left| x \right|+c$ | $\displaystyle \int{\frac{{{u}^{/}}}{u}dx=\ln \left| u \right|+c}$ | | $\displaystyle \int{\sin }xdx=-\cos x+C$ | $\displaystyle \int{{{u}^{/}}}\sin udx=-\cos u+c$ | | $\displaystyle \int{\cos x}dx=\sin x+c$ | $\displaystyle \int{{{u}^{/}}}\cos udx=\sin u+c$ | | $\displaystyle \int{\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\arcsin \frac{x}{a}}+c$ | $\displaystyle \int{\frac{{{u}^{/}}}{\sqrt{{{a}^{2}}-{{u}^{2}}}}dx=\arcsin \frac{u}{a}+c}$ | | $\displaystyle \int{\frac{1}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{\sqrt{{{u}^{2}}-{{a}^{2}}}}dx}=\ln \left| u+\sqrt{{{u}^{2}}-{{a}^{2}}} \right|+c$ | | $\displaystyle \int{\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}}dx=\ln \left( x+\sqrt{{{x}^{2}}+{{a}^{2}}} \right)+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{\sqrt{{{u}^{2}}+{{a}^{2}}}}dx=\ln \left( u+\sqrt{{{u}^{2}}+{{a}^{2}}} \right)+c}$ | | $\displaystyle \int{\frac{1}{{{x}^{2}}+{{a}^{2}}}dx=\frac{1}{a}arctg\frac{x}{a}+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{{{u}^{2}}+{{a}^{2}}}dx=\frac{1}{a}arctg\frac{u}{a}+c}$ | | $\displaystyle \int{\frac{1}{{{x}^{2}}-{{a}^{2}}}dx=\frac{1}{2a}\ln \left| \frac{x-a}{x+a} \right|+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{{{u}^{2}}-{{a}^{2}}}dx=\frac{1}{2a}\ln \left| \frac{u-a}{u+a} \right|+c}$ | | $\displaystyle \int{\frac{1}{{{x}^{2}}}dx=-\frac{1}{x}+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{{{u}^{2}}}dx=-\frac{1}{u}+c}$ | | $\displaystyle \int{\frac{1}{\sin x}dx=\ln \left| tg\frac{x}{2} \right|+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{\sin u}dx=\ln \left| tg\frac{u}{2} \right|+c}$ | | $\displaystyle \int{\frac{1}{{{\sin }^{2}}x}dx=-ctgx+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{{{\sin }^{2}}x}dx=-ctgu+c}$ | | $\displaystyle \int{\frac{1}{{{\cos }^{2}}x}dx=tgx+c}$ | $\displaystyle \int{\frac{{{u}^{/}}}{{{\cos }^{2}}u}dx=}tgu+c$ |

Tabelul Integralelor

Integrarea prin parti : $\displaystyle \int{f(x)g'(x)dx\text{=}f(x)g(x)\text{-}\int{f'(x)g(x)dx} }$

Primitive uzuale

$\displaystyle \int{\frac{\text{1}}{\text{ax+b}}dx=\frac{1}{a}\ln (ax+b)+C}$ $\displaystyle \int{\sqrt{x-a}dx=\frac{2}{3}{{(x-a)}^{3/2}}+C}$ $\displaystyle \int{\frac{\text{1}}{{{\text{(x+a)}}^{\text{2}}}}dx=\frac{-1}{x+a}+C}$ $\displaystyle \int{\frac{\text{1}}{\sqrt{\text{x}\pm \text{a}}}dx=2\sqrt{x\pm a}+C}$ $\displaystyle \int{{{\text{(x+a)}}^{\text{n}}}dx={{(x+a)}^{n}}(\frac{a}{1+n}+\frac{x}{1+n})+C},n\ne -1$ $\displaystyle \int{\frac{\text{1}}{\sqrt{a-x}}dx=2\sqrt{a-x}+C}$ $\displaystyle \int{\frac{\text{1}}{\text{(x+a)(x+b)}}dx=\frac{1}{b-a}\ln (\frac{x+a}{x+b})+C,a\ne b}$ $\displaystyle \int{\sqrt{ax+b}dx=(\frac{2b}{3a}+\frac{2x}{3})\sqrt{ax+b}+C}$ $\displaystyle \int{\frac{\text{x}}{\text{(x+a)(x+b)}}dx=\frac{1}{b-a}[a\ln (a+x)-b\ln (b+x)]+C,a\ne b}$ $\displaystyle \int{\frac{\text{x}}{\sqrt{\text{x}\pm \text{a}}}dx=\frac{2}{3}(x\pm 2a)\sqrt{x\pm a}+C}$ $\displaystyle \int{\frac{\text{x}}{{{\text{(x+a)}}^{\text{2}}}}dx=\frac{a}{x+a}+\ln (a+x)+C,a\ne b}$ $\displaystyle \int{{{e}^{ax}}dx=\frac{1}{a}{{e}^{ax}}+C}$ $\displaystyle \int{\sqrt{{{x}^{2}}\pm {{a}^{2}}}dx=\frac{1}{2}x\sqrt{{{x}^{2}}\pm {{a}^{2}}}\pm \frac{{{a}^{2}}}{2}\ln (x+\sqrt{{{x}^{2}}\pm {{a}^{2}}})+C}$ $\displaystyle \int{x{{e}^{x}}dx=(x-1){{e}^{x}}+C}$ $\displaystyle \int{\sqrt{{{a}^{2}}-{{x}^{2}}}dx=\frac{1}{2}x\sqrt{{{a}^{2}}-{{x}^{2}}}-\frac{{{a}^{2}}}{2}arctg(\frac{x\sqrt{{{a}^{2}}-{{x}^{2}}}}{{{x}^{2}}-{{a}^{2}}})+C}$ $\displaystyle \int{{{\sin }^{2}}(x)dx=\frac{x}{2}-\frac{1}{4}\sin (2x)+C}$ $\displaystyle \int{\text{ln(ax+b)}dx=\frac{ax+b}{a}\ln (ax+b)-x+C}$ $\displaystyle \int{{{\cos }^{2}}(x)dx=\frac{x}{2}+\frac{1}{4}\sin (2x)+C}$ $\displaystyle \int{\text{ln(x)}dx=x\ln (x)-x+C}$ $\displaystyle \int{\sin (x)\cos (x)dx=-\frac{1}{2}{{\cos }^{2}}(x)+C}$ $\displaystyle \int{{{e}^{x}}\sin (x)dx=\frac{1}{2}{{e}^{x}}[\sin (x)-\cos (x)]+C}$ $\displaystyle \int{x\cos (x)dx=\cos (x)+x\sin (x)+C}$ $\displaystyle \int{{{e}^{x}}co\operatorname{s}(x)dx=\frac{1}{2}{{e}^{x}}[\sin (x)+\cos (x)]+C}$ $\displaystyle \int{co\operatorname{s}(ax)dx=\frac{\sin (ax)}{a}+C}$ $\displaystyle \int{\frac{\text{dx}}{{{\text{(}{{\text{x}}^{\text{2}}}\text{+}{{\text{a}}^{\text{2}}}\text{)}}^{\text{2}}}}=\frac{x}{2{{a}^{3}}({{\text{x}}^{\text{2}}}\text{+}{{\text{a}}^{\text{2}}})}+\frac{1}{2{{a}^{3}}}arctg(\frac{x}{a})+C}$ $\displaystyle \int{\sin (ax)dx=-\frac{co\operatorname{s}(ax)}{a}+C}$ $\displaystyle \int{tg(x)dx=-\ln |\cos (x)|+C}$ $\displaystyle \int{ctg(x)dx=\ln |\sin (x)|+C}$ $\displaystyle \int{\frac{dx}{\sin (x)}=\ln |tg(\frac{x}{2})|+C}$ $\displaystyle \int{\frac{dx}{\cos (x)}=\ln |tg(\frac{\pi }{4}+\frac{x}{2})|+C}$