Integration Formulas (Table of Indefinite Integrals)
Below is a table of Indefinite Integrals. With this table and integration techniques, you will be able to find majority of integrals.
It is also worth noting that unlike derivative (we can find derivative of any function), we can't find integral of any function: this means that we can't find integral in terms of functions we know.
Examples of such functions are $$$\int{{e}}^{{{{x}}^{{2}}}}{d}{x}$$$, $$$\int\frac{{{\sin{{\left({x}\right)}}}}}{{x}}{d}{x}$$$, $$$\int\sqrt{{{{x}}^{{3}}+{1}}}{d}{x}$$$ etc.
| Basic Forms |
| $$$\int{\left({a}{f{{\left({x}\right)}}}+{b}{g{{\left({x}\right)}}}\right)}{d}{x}={a}\int{f{{\left({x}\right)}}}{d}{x}+{b}\int{g{{\left({x}\right)}}}{d}{x}$$$ where $$${a}$$$ and $$${b}$$$ are constants |
| $$$\int{u}{d}{v}={u}{v}-\int{v}{d}{u}$$$ (Integration by Parts) |
| $$$\int{{x}}^{{n}}{d}{x}={\left\{\begin{array}{c}\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}+{C}{\quad\text{if}\quad}{n}\ne-{1}\\{\ln}{\left|{x}\right|}+{C}{\quad\text{if}\quad}{n}=-{1}\\ \end{array}\right.}$$$ |
| $$$\int{{a}}^{{x}}{d}{x}=\frac{{{{a}}^{{x}}}}{{{\ln{{\left({a}\right)}}}}}+{C}$$$ |
| $$$\int{{e}}^{{x}}{d}{x}={{e}}^{{x}}+{C}$$$ |
| $$$\int{\sin{{\left({x}\right)}}}{d}{x}=-{\cos{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\cos{{\left({x}\right)}}}{d}{x}={\sin{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\sec}}^{{2}}{\left({x}\right)}{d}{x}={\tan{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\csc}}^{{2}}{\left({x}\right)}{d}{x}=-{\cot{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}{d}{x}={\sec{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}{d}{x}=-{\csc{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\tan{{\left({x}\right)}}}{d}{x}=-{\ln}{\left|{\cos{{\left({x}\right)}}}\right|}+{C}={\ln}{\left|{\sec{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{\cot{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\sin{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{\sec{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\sec{{\left({x}\right)}}}+{\tan{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{\csc{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\csc{{\left({x}\right)}}}-{\cot{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}={\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{a}}^{{2}}+{{x}}^{{2}}}}=\frac{{1}}{{a}}{\operatorname{arctan}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}=\frac{{1}}{{a}}\text{arcsec}{\left(\frac{{x}}{{a}}\right)}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{a}}^{{2}}-{{x}}^{{2}}}}=\frac{{1}}{{{2}{a}}}{\ln}{\left|\frac{{{x}+{a}}}{{{x}-{a}}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{x}}^{{2}}-{{a}}^{{2}}}}=\frac{{1}}{{{2}{a}}}{\ln}{\left|\frac{{{x}-{a}}}{{{x}+{a}}}\right|}+{C}$$$ |
| Exponential and Logarithmic Forms |
| $$$\int{x}{{e}}^{{{a}{x}}}{d}{x}=\frac{{1}}{{{{a}}^{{2}}}}{\left({a}{x}-{1}\right)}{{e}}^{{{a}{x}}}+{C}$$$ |
| $$$\int{{x}}^{{n}}{{e}}^{{{a}{x}}}{d}{x}=\frac{{1}}{{a}}{{x}}^{{n}}{{e}}^{{{a}{x}}}-\frac{{n}}{{a}}\int{{x}}^{{{n}-{1}}}{{e}}^{{{a}{x}}}{d}{x}$$$ |
| $$$\int{{e}}^{{{a}{x}}}{\sin{{\left({b}{x}\right)}}}{d}{x}=\frac{{{{e}}^{{{a}{x}}}}}{{{{a}}^{{2}}+{{b}}^{{2}}}}{\left({\operatorname{asin}{{\left({b}{x}\right)}}}-{b}{\cos{{\left({b}{x}\right)}}}\right)}+{C}$$$ |
| $$$\int{{e}}^{{{a}{x}}}{\cos{{\left({b}{x}\right)}}}{d}{x}=\frac{{{{e}}^{{{a}{x}}}}}{{{{a}}^{{2}}+{{b}}^{{2}}}}{\left({\operatorname{acos}{{\left({b}{x}\right)}}}+{b}{\sin{{\left({b}{x}\right)}}}\right)}+{C}$$$ |
| $$$\int{\ln{{\left({x}\right)}}}{d}{x}={x}{\left({\ln{{\left({x}\right)}}}-{1}\right)}+{C}$$$ |
| $$$\int{{x}}^{{n}}{\ln{{\left({x}\right)}}}{d}{x}=\frac{{{{x}}^{{{n}+{1}}}}}{{{{\left({n}+{1}\right)}}^{{2}}}}{\left({\left({n}+{1}\right)}{\ln{{\left({x}\right)}}}-{1}\right)}+{C}$$$ |
| $$$\int\frac{{1}}{{{x}{\ln{{\left({x}\right)}}}}}{d}{x}={\ln}{\left|{\ln{{\left({u}\right)}}}\right|}+{C}$$$ |
| Trigonometric Forms |
| $$$\int{{\sin}}^{{2}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{x}-\frac{{1}}{{4}}{\sin{{\left({2}{x}\right)}}}+{C}$$$ |
| $$$\int{{\cos}}^{{2}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{x}+\frac{{1}}{{4}}{\sin{{\left({2}{x}\right)}}}+{C}$$$ |
| $$$\int{{\tan}}^{{2}}{\left({x}\right)}{d}{x}={\tan{{\left({x}\right)}}}-{x}+{C}$$$ |
| $$$\int{{\cot}}^{{2}}{\left({x}\right)}{d}{x}=-{\cot{{\left({x}\right)}}}-{x}+{C}$$$ |
| $$$\int{{\sin}}^{{3}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{3}}{\left({2}+{{\sin}}^{{2}}{\left({x}\right)}\right)}{\cos{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\cos}}^{{3}}{\left({x}\right)}{d}{x}=\frac{{1}}{{3}}{\left({2}+{{\cos}}^{{2}}{\left({x}\right)}\right)}{\sin{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\tan}}^{{3}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{{\tan}}^{{2}}{\left({x}\right)}+{\ln}{\left|{\cos{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{{\cot}}^{{3}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{2}}{{\cot}}^{{2}}{\left({x}\right)}-{\ln}{\left|{\sin{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{{\sec}}^{{3}}{\left({x}\right)}{d}{x}=\frac{{1}}{{2}}{\sec{{\left({x}\right)}}}{\tan{{\left({x}\right)}}}+\frac{{1}}{{2}}{\ln}{\left|{\sec{{\left({x}\right)}}}+{\tan{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{{\csc}}^{{3}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{2}}{\csc{{\left({x}\right)}}}{\cot{{\left({x}\right)}}}+\frac{{1}}{{2}}{\ln}{\left|{\csc{{\left({x}\right)}}}-{\cot{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{{\sin}}^{{n}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{n}}{{\sin}}^{{{n}-{1}}}{\left({x}\right)}{\cos{{\left({x}\right)}}}+\frac{{{n}-{1}}}{{n}}\int{{\sin}}^{{{n}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| $$$\int{{\cos}}^{{n}}{\left({x}\right)}{d}{x}=\frac{{1}}{{n}}{{\cos}}^{{{n}-{1}}}{\left({x}\right)}{\sin{{\left({x}\right)}}}+\frac{{{n}-{1}}}{{n}}\int{{\cos}}^{{{n}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| $$$\int{{\tan}}^{{n}}{\left({x}\right)}{d}{x}=\frac{{1}}{{{n}-{1}}}{{\tan}}^{{{n}-{1}}}{\left({x}\right)}-\int{{\tan}}^{{{n}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| $$$\int{{\cot}}^{{n}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{{n}-{1}}}{{\cot}}^{{{n}-{1}}}{\left({x}\right)}-\int{{\cot}}^{{{n}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| $$$\int{{\sec}}^{{n}}{\left({x}\right)}{d}{x}=\frac{{1}}{{{n}-{1}}}{\tan{{\left({x}\right)}}}{{\sec}}^{{{n}-{2}}}{\left({x}\right)}+\frac{{{n}-{2}}}{{{n}-{1}}}\int{{\sec}}^{{{n}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| $$$\int{{\csc}}^{{n}}{\left({x}\right)}{d}{x}=-\frac{{1}}{{{n}-{1}}}{\cot{{\left({x}\right)}}}{{\csc}}^{{{n}-{2}}}{\left({x}\right)}+\frac{{{n}-{2}}}{{{n}-{1}}}\int{{\csc}}^{{{n}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| $$$\int{\sin{{\left({a}{x}\right)}}}{\sin{{\left({b}{x}\right)}}}{d}{x}=\frac{{{\sin{{\left({\left({a}-{b}\right)}{x}\right)}}}}}{{{2}{\left({a}-{b}\right)}}}-\frac{{{\sin{{\left({\left({a}+{b}\right)}{x}\right)}}}}}{{{2}{\left({a}+{b}\right)}}}+{C}$$$ |
| $$$\int{\cos{{\left({a}{x}\right)}}}{\cos{{\left({b}{x}\right)}}}{d}{x}=\frac{{{\sin{{\left({\left({a}-{b}\right)}{x}\right)}}}}}{{{2}{\left({a}-{b}\right)}}}+\frac{{{\sin{{\left({\left({a}+{b}\right)}{x}\right)}}}}}{{{2}{\left({a}+{b}\right)}}}+{C}$$$ |
| $$$\int{\sin{{\left({a}{x}\right)}}}{\cos{{\left({b}{x}\right)}}}{d}{x}=-\frac{{{\cos{{\left({\left({a}-{b}\right)}{x}\right)}}}}}{{{2}{\left({a}-{b}\right)}}}-\frac{{{\cos{{\left({\left({a}+{b}\right)}{x}\right)}}}}}{{{2}{\left({a}+{b}\right)}}}+{C}$$$ |
| $$$\int{x}{\sin{{\left({x}\right)}}}{d}{x}={\sin{{\left({x}\right)}}}-{x}{\cos{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{x}{\cos{{\left({x}\right)}}}{d}{x}={\cos{{\left({x}\right)}}}+{x}{\sin{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{x}}^{{n}}{\sin{{\left({x}\right)}}}{d}{x}=-{{x}}^{{n}}{\cos{{\left({x}\right)}}}+{n}\int{{x}}^{{{n}-{1}}}{\cos{{\left({x}\right)}}}{d}{x}$$$ |
| $$$\int{{x}}^{{n}}{\cos{{\left({x}\right)}}}{d}{x}={{x}}^{{n}}{\sin{{\left({x}\right)}}}-{n}\int{{x}}^{{{n}-{1}}}{\sin{{\left({x}\right)}}}{d}{x}$$$ |
| $$$\int{{\sin}}^{{n}}{\left({x}\right)}{{\cos}}^{{m}}{\left({x}\right)}{d}{x}=-\frac{{{{\sin}}^{{{n}-{1}}}{\left({x}\right)}{{\cos}}^{{{m}+{1}}}{\left({x}\right)}}}{{{n}+{m}}}+\frac{{{n}-{1}}}{{{n}+{m}}}\int{{\sin}}^{{{n}-{2}}}{\left({x}\right)}{{\cos}}^{{m}}{\left({x}\right)}{d}{x}=$$$ $$$=\frac{{{{\sin}}^{{{n}+{1}}}{\left({x}\right)}{{\cos}}^{{{m}-{1}}}{\left({x}\right)}}}{{{n}+{m}}}+\frac{{{m}-{1}}}{{{n}+{m}}}\int{{\sin}}^{{n}}{\left({x}\right)}{{\cos}}^{{{m}-{2}}}{\left({x}\right)}{d}{x}$$$ |
| Inverse Trigonometric Forms |
| $$$\int{\operatorname{arcsin}{{\left({x}\right)}}}{d}{x}={x}{\operatorname{arcsin}{{\left({x}\right)}}}+\sqrt{{{1}-{{x}}^{{2}}}}+{C}$$$ |
| $$$\int{\operatorname{arccos}{{\left({x}\right)}}}{d}{x}={x}{\operatorname{arccos}{{\left({x}\right)}}}-\sqrt{{{1}-{{x}}^{{2}}}}+{C}$$$ |
| $$$\int{\operatorname{arctan}{{\left({x}\right)}}}{d}{x}={x}{\operatorname{arctan}{{\left({x}\right)}}}-\frac{{1}}{{2}}{\ln{{\left({1}+{{x}}^{{2}}\right)}}}+{C}$$$ |
| $$$\int{x}{\operatorname{arcsin}{{\left({x}\right)}}}{d}{x}=\frac{{{2}{{x}}^{{2}}-{1}}}{{4}}{\operatorname{arcsin}{{\left({x}\right)}}}+\frac{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}{{4}}+{C}$$$ |
| $$$\int{x}{\operatorname{arccos}{{\left({x}\right)}}}{d}{x}=\frac{{{2}{{x}}^{{2}}-{1}}}{{4}}{\operatorname{arccos}{{\left({x}\right)}}}-\frac{{{x}\sqrt{{{1}-{{x}}^{{2}}}}}}{{4}}+{C}$$$ |
| $$$\int{x}{\operatorname{arctan}{{\left({x}\right)}}}{d}{x}=\frac{{{{x}}^{{2}}+{1}}}{{2}}{\operatorname{arctan}{{\left({x}\right)}}}-\frac{{x}}{{2}}+{C}$$$ |
| $$$\int{{x}}^{{n}}{\operatorname{arcsin}{{\left({x}\right)}}}{d}{x}=\frac{{1}}{{{n}+{1}}}{\left({{x}}^{{{n}+{1}}}{\operatorname{arcsin}{{\left({x}\right)}}}-\int\frac{{{{x}}^{{{n}+{1}}}}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}{d}{x}\right)}$$$, $$${n}\ne-{1}$$$ |
| $$$\int{{x}}^{{n}}{\operatorname{arccos}{{\left({x}\right)}}}{d}{x}=\frac{{1}}{{{n}+{1}}}{\left({{x}}^{{{n}+{1}}}{\operatorname{arccos}{{\left({x}\right)}}}+\int\frac{{{{x}}^{{{n}+{1}}}}}{{\sqrt{{{1}-{{x}}^{{2}}}}}}{d}{x}\right)}$$$, $$${n}\ne-{1}$$$ |
| $$$\int{{x}}^{{n}}{\operatorname{arctan}{{\left({x}\right)}}}{d}{x}=\frac{{1}}{{{n}+{1}}}{\left({{x}}^{{{n}+{1}}}{\operatorname{arctan}{{\left({x}\right)}}}-\int\frac{{{{x}}^{{{n}+{1}}}}}{{{1}+{{x}}^{{2}}}}{d}{x}\right)}$$$, $$${n}\ne-{1}$$$ |
| Hyperbolic Forms |
| $$$\int{\sinh{{\left({x}\right)}}}{d}{x}={\cosh{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\cosh{{\left({x}\right)}}}{d}{x}={\sinh{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\tanh{{\left({x}\right)}}}{d}{x}={\ln{{\left({\cosh{{\left({x}\right)}}}\right)}}}+{C}$$$ |
| $$$\int{\coth{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\sinh{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{\operatorname{sech}{{\left({x}\right)}}}{d}{x}={\operatorname{arctan}}{\left|{\sinh{{\left({x}\right)}}}\right|}+{C}$$$ |
| $$$\int{\operatorname{csch}{{\left({x}\right)}}}{d}{x}={\ln}{\left|{\tanh{{\left(\frac{{1}}{{2}}{x}\right)}}}\right|}+{C}$$$ |
| $$$\int{{\operatorname{sech}}}^{{2}}{\left({x}\right)}{d}{x}={\tanh{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{{\operatorname{csch}}}^{{2}}{\left({x}\right)}{d}{x}=-{\coth{{\left({x}\right)}}}+{C}$$$ |
| $$$\int{\operatorname{sech}{{\left({x}\right)}}}{\tanh{{\left({x}\right)}}}{d}{x}=-{s}{e}{c}{h}{\left({x}\right)}+{C}$$$ |
| $$$\int{\operatorname{csch}{{\left({x}\right)}}}{\coth{{\left({x}\right)}}}{d}{x}=-{c}{s}{c}{h}{\left({x}\right)}+{C}$$$ |
| Forms Involving $$$\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}$$$, $$${a}>{0}$$$ |
| $$$\int\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C}$$$ |
| $$$\int{{x}}^{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{8}}{\left({{a}}^{{2}}+{2}{{x}}^{{2}}\right)}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}-\frac{{{{a}}^{{4}}}}{{8}}{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{x}}{d}{x}=\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}-{a}{\ln}{\left|\frac{{{a}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{x}}\right|}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{x}}+{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}={\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}{d}{x}}}{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}=\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}-\frac{{{{a}}^{{2}}}}{{2}}{\ln{{\left({x}+\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}=-\frac{{1}}{{a}}{\ln}{\left|\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}+{a}}}{{x}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{x}}^{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}=-\frac{{\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}{{{{a}}^{{2}}{x}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{\left({{a}}^{{2}}+{{x}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}}}=\frac{{x}}{{{{a}}^{{2}}\sqrt{{{{a}}^{{2}}+{{x}}^{{2}}}}}}+{C}$$$ |
| Forms Involving $$$\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}$$$, $$${a}>{0}$$$ |
| $$$\int\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int{{x}}^{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}{d}{x}=\frac{{x}}{{8}}{\left({2}{{x}}^{{2}}-{{a}}^{{2}}\right)}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{{a}}^{{4}}}}{{8}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{x}}=\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}-{a}{\ln}{\left|\frac{{{a}+\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{x}}\right|}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{1}}{{x}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}-{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}}}{\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}{d}{x}=-\frac{{x}}{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}=-\frac{{1}}{{a}}{\ln}{\left|\frac{{{a}+\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}{{x}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{x}}^{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}=-\frac{{1}}{{{{a}}^{{2}}{x}}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+{C}$$$ |
| $$$\int{{\left({{a}}^{{2}}-{{x}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}{d}{x}=-\frac{{x}}{{8}}{\left({2}{{x}}^{{2}}-{5}{{a}}^{{2}}\right)}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}+\frac{{{3}{{a}}^{{4}}}}{{8}}{\operatorname{arcsin}{{\left(\frac{{x}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{\left({{a}}^{{2}}-{{x}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}}}=\frac{{x}}{{{{a}}^{{2}}\sqrt{{{{a}}^{{2}}-{{x}}^{{2}}}}}}+{C}$$$ |
| Forms Involving $$$\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}$$$, $$${a}>{0}$$$ |
| $$$\int\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}-\frac{{{{a}}^{{2}}}}{{2}}{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C}$$$ |
| $$$\int{{x}}^{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}{d}{x}=\frac{{x}}{{8}}{\left({2}{{x}}^{{2}}-{{a}}^{{2}}\right)}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}-\frac{{{{a}}^{{4}}}}{{8}}{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{x}}{d}{x}=\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}-{a}\cdot{\operatorname{arccos}{{\left(\frac{{a}}{{{\left|{u}\right|}}}\right)}}}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{x}}+{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}={\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}}}{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{d}{x}=\frac{{x}}{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\ln}{\left|{x}+\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{x}}^{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}=\frac{{\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}{{{{a}}^{{2}}{x}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{\left({{x}}^{{2}}-{{a}}^{{2}}\right)}}^{{\frac{{3}}{{2}}}}}}=-\frac{{x}}{{{{a}}^{{2}}\sqrt{{{{x}}^{{2}}-{{a}}^{{2}}}}}}+{C}$$$ |
| Forms Involving $$${a}+{b}{x}$$$ |
| $$$\int\frac{{x}}{{{a}+{b}{x}}}{d}{x}=\frac{{1}}{{{{b}}^{{2}}}}{\left({a}+{b}{x}-{a}{\ln}{\left|{a}+{b}{x}\right|}\right)}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}}}{{{a}+{b}{x}}}{d}{x}=\frac{{1}}{{{2}{{b}}^{{3}}}}{\left({{\left({a}+{b}{x}\right)}}^{{2}}-{4}{a}{\left({a}+{b}{x}\right)}+{2}{{a}}^{{2}}{\ln}{\left|{a}+{b}{x}\right|}\right)}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}{\left({a}+{b}{x}\right)}}}=\frac{{1}}{{a}}{\ln}{\left|\frac{{x}}{{{a}+{b}{x}}}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{x}}^{{2}}{\left({a}+{b}{x}\right)}}}=-\frac{{1}}{{{a}{x}}}+\frac{{b}}{{{{a}}^{{2}}}}{\ln}{\left|\frac{{{a}+{b}{x}}}{{x}}\right|}+{C}$$$ |
| $$$\int\frac{{x}}{{{{\left({a}+{b}{x}\right)}}^{{2}}}}{d}{x}=\frac{{a}}{{{{b}}^{{2}}{\left({a}+{b}{x}\right)}}}+\frac{{1}}{{{{b}}^{{2}}}}{\ln}{\left|{a}+{b}{x}\right|}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}{{\left({a}+{b}{x}\right)}}^{{2}}}}=\frac{{1}}{{{a}{\left({a}+{b}{x}\right)}}}-\frac{{1}}{{{{a}}^{{2}}}}{\ln}{\left|\frac{{{a}+{b}{x}}}{{x}}\right|}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}}}{{{{\left({a}+{b}{x}\right)}}^{{2}}}}{d}{x}=\frac{{1}}{{{{b}}^{{3}}}}{\left({a}+{b}{x}-\frac{{{{a}}^{{2}}}}{{{a}+{b}{x}}}-{2}{a}{\ln}{\left|{a}+{b}{x}\right|}\right)}+{C}$$$ |
| $$$\int{x}\sqrt{{{a}+{b}{x}}}{d}{x}=\frac{{2}}{{{15}{{b}}^{{2}}}}{\left({3}{b}{x}-{2}{a}\right)}{{\left({a}+{b}{x}\right)}}^{{\frac{{3}}{{2}}}}+{C}$$$ |
| $$$\int\frac{{x}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}=\frac{{2}}{{{3}{{b}}^{{2}}}}{\left({b}{x}-{2}{a}\right)}\sqrt{{{a}+{b}{x}}}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}=\frac{{2}}{{{15}{{b}}^{{3}}}}{\left({8}{{a}}^{{2}}+{3}{{b}}^{{2}}{{x}}^{{2}}-{4}{a}{b}{x}\right)}\sqrt{{{a}+{b}{x}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}\sqrt{{{a}+{b}{x}}}}}={\left\{\begin{array}{c}\frac{{1}}{{\sqrt{{{a}}}}}{\ln}{\left|\frac{{\sqrt{{{a}+{b}{x}}}-\sqrt{{{a}}}}}{{\sqrt{{{a}+{b}{x}}}+\sqrt{{{a}}}}}\right|}+{C}{\quad\text{if}\quad}{a}>{0}\\\frac{{2}}{{\sqrt{{-{a}}}}}{\operatorname{arctan}{{\left(\sqrt{{\frac{{{a}+{b}{x}}}{{-{a}}}}}\right)}}}+{C}{\quad\text{if}\quad}{a}<{0}\\ \end{array}\right.}$$$ |
| $$$\int\frac{{\sqrt{{{a}+{b}{x}}}}}{{x}}{d}{x}={2}\sqrt{{{a}+{b}{x}}}+{a}\int\frac{{{d}{x}}}{{{x}\sqrt{{{a}+{b}{x}}}}}$$$ |
| $$$\int\frac{{\sqrt{{{a}+{b}{x}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{\sqrt{{{a}+{b}{x}}}}}{{x}}+\frac{{b}}{{2}}\int\frac{{{d}{x}}}{{{x}\sqrt{{{a}+{b}{x}}}}}$$$ |
| $$$\int{{x}}^{{n}}\sqrt{{{a}+{b}{x}}}{d}{x}=\frac{{2}}{{{b}{\left({2}{n}+{3}\right)}}}{\left({{x}}^{{n}}{{\left({a}+{b}{x}\right)}}^{{\frac{{3}}{{2}}}}-{n}{a}\int{{x}}^{{{n}-{1}}}\sqrt{{{a}+{b}{x}}}{d}{x}\right)}$$$ |
| $$$\int\frac{{{{x}}^{{n}}}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}=\frac{{{2}{{x}}^{{n}}\sqrt{{{a}+{b}{x}}}}}{{{b}{\left({2}{n}+{1}\right)}}}-\frac{{{2}{n}{a}}}{{{b}{\left({2}{n}+{1}\right)}}}\int\frac{{{{x}}^{{{n}-{1}}}}}{{\sqrt{{{a}+{b}{x}}}}}{d}{x}$$$ |
| $$$\int\frac{{{d}{x}}}{{{{x}}^{{n}}\sqrt{{{a}+{b}{x}}}}}=-\frac{{\sqrt{{{a}+{b}{x}}}}}{{{a}{\left({n}-{1}\right)}{{x}}^{{{n}-{1}}}}}-\frac{{{b}{\left({2}{n}-{3}\right)}}}{{{2}{a}{\left({n}-{1}\right)}}}\int\frac{{{d}{x}}}{{{{x}}^{{{n}-{1}}}\sqrt{{{a}+{b}{x}}}}}$$$ |
| Forms Involving $$$\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}$$$, $$${a}>{0}$$$ |
| $$$\int\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}{d}{x}=\frac{{{x}-{a}}}{{2}}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+\frac{{{{a}}^{{2}}}}{{2}}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int{x}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}{d}{x}=\frac{{{2}{{x}}^{{2}}-{a}{x}-{3}{{a}}^{{2}}}}{{6}}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+\frac{{{{a}}^{{3}}}}{{2}}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{x}}{d}{x}=\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+{a}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{{{x}}^{{2}}}}{d}{x}=-\frac{{{2}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{x}}-{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}={\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{x}}{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{d}{x}=-\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+{a}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{{x}}^{{2}}}}{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{d}{x}=-\frac{{{x}+{3}{a}}}{{2}}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}+\frac{{{3}{{a}}^{{2}}}}{{2}}{\operatorname{arccos}{{\left(\frac{{{a}-{x}}}{{a}}\right)}}}+{C}$$$ |
| $$$\int\frac{{{d}{x}}}{{{x}\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}=-\frac{{\sqrt{{{2}{a}{x}-{{x}}^{{2}}}}}}{{{a}{x}}}+{C}$$$ |