| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| 1 | x | \frac{1}{2} x^2 + C |
| 0 | a | ax + C |
| 1 | x \pm a | \frac{1}{2} x^2 \pm ax + C |
| a | ax | \frac{1}{2} ax^2 + C |
| 2x | x^2 | \frac{1}{3} x^3 + C |
| nx^{n-1} | x^n | \dfrac{1}{n+1} x^{n+1} + C |
| -x^{-2} | x^{-1} | \ln |x| + C |
| \dfrac{du}{dx} \pm \dfrac{dv}{dx} \pm \dfrac{dw}{dx} | u \pm v \pm w | \int u\, dx \pm \int v\, dx \pm \int w\, dx |
| u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx} | uv | Nenhuma forma geral conhecida |
| \dfrac{v\, \dfrac{du}{dx} - u\, \dfrac{dv}{dx}}{v^2} | \dfrac{u}{v} | Nenhuma forma geral conhecida |
| \dfrac{du}{dx} | u | ux - \int x\, du + C |
Exponencial e Logarítmica:
| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| e^x | e^x | e^x + C |
| x^{-1} | \ln x | x(\ln x - 1) + C |
| \dfrac{1}{x\,\ln b} | \log_{b} x | \dfrac{1}{\ln b} x (\ln x - 1) + C |
| a^x \ln a | a^x | \dfrac{a^x}{\ln a} + C |
Trigonométrica:
| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| \cos x | \sin x | -\cos x + C |
| -\sin x | \cos x | \sin x + C |
| \sec^2 x | \tan x | -\ln|\cos x| + C |
| \sec x\ \tan x | \sec x | \ln|\sec x+\tan x| + C |
Trigonométrica Inversa:
| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| \dfrac{1}{\sqrt{1-x^2}} | \arcsin x=\sin^{-1} x | x \cdot \arcsin x + \sqrt{1 - x^2} + C |
| -\dfrac{1}{\sqrt{1-x^2}} | \arccos x=\cos^{-1} x | x \cdot \arccos x - \sqrt{1 - x^2} + C |
| \dfrac{1}{1+x^2} | \arctan x=\tan^{-1} x | x \cdot \arctan x - \frac{1}{2} \ln (1 + x^2) + C |
Hiperbólica:
| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| \cosh x | \sinh x | \cosh x + C |
| \sinh x | \cosh x | \sinh x + C |
| \text{sech}^2 x | \tanh x | \ln \cosh x + C |
| -\text{sech } x \tanh x | \text{sech }x | 2\arctan\left(e^x\right)+C |
Hiperbólica Inversa:
| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| \dfrac{1}{\sqrt{x^2+1}} | \text{arcsinh } x=\sinh^{-1} x | -\sqrt{1 + x^2} + x \text{ arcsinh }x+C |
| \dfrac{1}{\sqrt{x^2-1}} | \text{arccosh } x=\cosh^{-1} x | -\sqrt{x^2-1}+x \text{ arccosh }x+C |
| \dfrac{1}{1-x^2} | \text{arctanh }x = \tanh^{-1} x | x \text{ arctanh } x + \dfrac{1}{2} \ln|1 - x^2|+C |
Diversos:
| \dfrac{dy}{dx} | \longleftarrow\quad y\quad\longrightarrow | \int y\, dx |
|---|---|---|
| -\dfrac{1}{(x + a)^2} | \dfrac{1}{x + a} | \ln |x+a| + C |
| \dfrac{x}{\sqrt{x^2+a^2}} | \sqrt{a^2+x^2} | \dfrac{1}{2}x\sqrt{x^2+a^2}+\dfrac{a^2}{2}\ln\left(x+\sqrt{x^2+a^2}\right)+C |
| -\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}} | \dfrac{1}{\sqrt{a^2 + x^2}} | \ln (x + \sqrt{a^2 + x^2}) + C |
| \mp \dfrac{b}{(a \pm bx)^2} | \dfrac{1}{a \pm bx} | \pm \dfrac{1}{b} \ln |a \pm bx|+ C |
| -\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}} | \dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}} | \dfrac{x}{\sqrt{a^2 + x^2}} + C |
| a \cdot \cos ax | \sin ax | -\dfrac{1}{a} \cos ax + C |
| -a \cdot \sin ax | \cos ax | \dfrac{1}{a} \sin ax + C |
| a \cdot \sec^2ax | \tan ax | -\dfrac{1}{a} \ln |\cos ax| + C |
| \sin 2x | \sin^2 x | \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C |
| -\sin 2x | \cos^2 x | \dfrac{x}{2} + \dfrac{\sin 2x}{4} + C |
| n \cdot \sin^{n-1} x \cdot \cos x | \sin^n x | -\dfrac{\cos x}{n} \sin^{n-1} x + \dfrac{n-1}{n} \int \sin^{n-2} x\, dx |
| -\dfrac{\cos x}{\sin^2 x} | \dfrac{1}{\sin x} | \ln\left|\tan \dfrac{x}{2}\right| + C |
| -\dfrac{\sin 2x}{\sin^4 x} | \dfrac{1}{\sin^2 x} | -\cot x + C |
| \dfrac{\sin^2 x - \cos^2 x}{\sin^2 x \cdot \cos^2 x} | \dfrac{1}{\sin x \cdot \cos x} | \ln|\tan x| + C |
| n \cdot \sin mx \cdot \cos nx + m \cdot \sin nx \cdot \cos mx | \sin mx \cdot \sin nx | \frac{1}{2} \cos(m - n)x - \frac{1}{2} \cos(m + n)x + C |
| 2a\cdot\sin 2ax | \sin^2 ax | \dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C |
| -2a\cdot\sin 2ax | \cos^2 ax | \dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C |
