| \(\dfrac{dy}{dx}\) ⬅️⬅️⬅️⬅️⬅️ | \( y \) | ➡️➡️➡️➡️➡️➡️ \(\displaystyle \int y\, dx\) |
Algebraic. |
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| \(1\) | \(x\) | \(\frac{1}{2} x^2 + C\) |
| \(0\) | \(a\) | \(ax + C\) |
| \(1\) | \(x \pm a\) | \(\frac{1}{2} x^2 \pm ax + C\) |
| \(a\) | \(ax\) | \(\frac{1}{2} ax^2 + C\) |
| \(2x\) | \(x^2\) | \(\frac{1}{3} x^3 + C\) |
| \(nx^{n-1}\) | \(x^n\) | \(\dfrac{1}{n+1} x^{n+1} + C\) |
| \(-x^{-2}\) | \(x^{-1}\) | \(\ln x + C\) |
| \(\dfrac{du}{dx} \pm \dfrac{dv}{dx} \pm \dfrac{dw}{dx}\) | \(u \pm v \pm w\) | \(\displaystyle \int u\, dx \pm \int v\, dx \pm \int w\, dx\) |
| \(u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}\) | \(uv\) | No general form known |
| \(\dfrac{v\, \dfrac{du}{dx} – u\, \dfrac{dv}{dx}}{v^2}\) | \(\dfrac{u}{v}\) | No general form known |
| \(\dfrac{du}{dx}\) | \(u\) | \(\displaystyle ux – \int x\, du + C\) |
Exponential and Logarithmic. |
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| \(e^x\) | \(e^x\) | \(e^x + C\) |
| \(x^{-1}\) | \(\ln x\) | \(x(\ln x – 1) + C\) |
| \(0.4343 \times x^{-1}\) | \(\log_{10} x\) | \(0.4343x (\ln x – 1) + C\) |
| \(a^x \ln a\) | \(a^x\) | \(\dfrac{a^x}{\ln a} + C\) |
Trigonometrical. |
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| \(\cos x\) | \(\sin x\) | \(-\cos x + C\) |
| \(-\sin x\) | \(\cos x\) | \(\sin x + C\) |
| \(\sec^2 x\) | \(\tan x\) | \(-\ln \cos x + C\) |
Circular (Inverse). |
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| \(\dfrac{1}{\sqrt{(1-x^2)}}\) | \(\arcsin x\) | \(x \cdot \arcsin x + \sqrt{1 – x^2} + C\) |
| \(-\dfrac{1}{\sqrt{(1-x^2)}}\) | \(\arccos x\) | \(x \cdot \arccos x – \sqrt{1 – x^2} + C\) |
| \(\dfrac{1}{1+x^2}\) | \(\arctan x\) | \(x \cdot \arctan x – \frac{1}{2} \log_\epsilon (1 + x^2) + C\) |
Hyperbolic. |
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| \(\cosh x\) | \(\sinh x\) | \(\cosh x + C\) |
| \(\sinh x\) | \(\cosh x\) | \(\sinh x + C\) |
| \(\text{sech}^2 x\) | \(\tanh x\) | \(\ln \cosh x + C\) |
| \(e^x\) | \(e^x\) | \(e^x + C\) |
| \(x^{-1}\) | \(\ln x\) | \(x(\ln x – 1) + C\) |
| \(0.4343 \times x^{-1}\) | \(\log_{10} x\) | \(0.4343x (\ln x – 1) + C\) |
| \(a^x \ln a\) | \(a^x\) | \(\dfrac{a^x}{\ln a} + C\) |
Miscellaneous. |
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| \(-\dfrac{1}{(x + a)^2}\) |
$\dfrac{1}{x + a}$ |
$\ln (x+a) + C$ |
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$-\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}}$ |
\(\dfrac{1}{\sqrt{a^2 + x^2}}\) |
\(\ln (x + \sqrt{a^2 + x^2}) + C\) |
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\(\mp \dfrac{b}{(a \pm bx)^2}\) |
\(\dfrac{1}{a \pm bx}\) |
\(\pm \dfrac{1}{b} \ln (a \pm bx) + C\) |
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\(-\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}}\) |
\(\dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}}\) |
\(\dfrac{x}{\sqrt{a^2 + x^2}} + C\) |
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\(a \cdot \cos ax\) |
\(\sin ax\) |
\(-\dfrac{1}{a} \cos ax + C\) |
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\(-a \cdot \sin ax\) |
\(\cos ax\) |
\(\dfrac{1}{a} \sin ax + C\) |
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\(a \cdot \sec^2ax\) |
\(\tan ax\) |
\(-\dfrac{1}{a} \ln \cos ax + C\) |
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\(\sin 2x\) |
\(\sin^2 x\) |
\(\dfrac{x}{2} – \dfrac{\sin 2x}{4} + C\) |
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\(-\sin 2x\) |
\(\cos^2 x\) |
$\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C $ |
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$ 2a\cdot\sin 2ax$ |
$\sin^2 ax$ |
$\dfrac{x}{2} – \dfrac{\sin 2ax}{4a} + C $ |
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$-2a\cdot\sin 2ax$ |
$\cos^2 ax$ |
$\dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C $ |