\(\dfrac{dy}{dx}\) \( \leftarrow y \rightarrow\) \(\int y\, dx\)
 

Algebraic.

 
\(1\) \(x\) \(\frac{1}{2} x^2 + C\)
\(0\) \(a\) \(ax + C\)
\(1\) \(x \pm a\) \(\frac{1}{2} x^2 \pm ax + C\)
\(a\) \(ax\) \(\frac{1}{2} ax^2 + C\)
\(2x\) \(x^2\) \(\frac{1}{3} x^3 + C\)
\(nx^{n-1}\) \(x^n\) \(\dfrac{1}{n+1} x^{n+1} + C\)
\(-x^{-2}\) \(x^{-1}\) \(\log_\epsilon x + C\)
\(\dfrac{du}{dx} \pm \dfrac{dv}{dx} \pm \dfrac{dw}{dx}\) \(u \pm v \pm w\) \(\int u\, dx \pm \int v\, dx \pm \int w\, dx\)
\(u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}\) \(uv\) No general form known
\(\dfrac{v\, \dfrac{du}{dx} – u\, \dfrac{dv}{dx}}{v^2}\) \(\dfrac{u}{v}\) No general form known
\(\dfrac{du}{dx}\) \(u\) \(ux – \int x\, du + C\)
 

Exponential and Logarithmic.

 
\(\epsilon^x\) \(\epsilon^x\) \(\epsilon^x + C\)
\(x^{-1}\) \(\log_\epsilon x\) \(x(\log_\epsilon x – 1) + C\)
\(0.4343 \times x^{-1}\) \(\log_{10} x\) \(0.4343x (\log_\epsilon x – 1) + C\)
\(a^x \log_\epsilon a\) \(a^x\) \(\dfrac{a^x}{\log_\epsilon a} + C\)
 

Trigonometrical.

 
\(\cos x\) \(\sin x\) \(-\cos x + C\)
\(-\sin x\) \(\cos x\) \(\sin x + C\)
\(\sec^2 x\) \(\tan x\) \(-\log_\epsilon \cos x + C\)
 

Circular (Inverse).

 
\(\dfrac{1}{\sqrt{(1-x^2)}}\) \(\arcsin x\) \(x \cdot \arcsin x + \sqrt{1 – x^2} + C\)
\(-\dfrac{1}{\sqrt{(1-x^2)}}\) \(\arccos x\) \(x \cdot \arccos x – \sqrt{1 – x^2} + C\)
\(\dfrac{1}{1+x^2}\) \(\arctan x\) \(x \cdot \arctan x – \frac{1}{2} \log_\epsilon (1 + x^2) + C\)
 

Hyperbolic.

 
\(\cosh x\) \(\sinh x\) \(\cosh x + C\)
\(\sinh x\) \(\cosh x\) \(\sinh x + C\)
\(\text{sech}^2 x\) \(\tanh x\) \(\log_\epsilon \cosh x + C\)
\(\epsilon^x\) \(\epsilon^x\) \(\epsilon^x + C\)
\(x^{-1}\) \(\log_\epsilon x\) \(x(\log_\epsilon x – 1) + C\)
\(0.4343 \times x^{-1}\) \(\log_{10} x\) \(0.4343x (\log_\epsilon x – 1) + C\)
\(a^x \log_\epsilon a\) \(a^x\) \(\dfrac{a^x}{\log_\epsilon a} + C\)
 

Miscellaneous.

 
\(-\dfrac{1}{(x + a)^2}\)

$\dfrac{1}{x + a}$

$\log_\epsilon (x+a) + C$

$-\dfrac{x}{(a^2 + x^2)^{\frac{3}{2}}}$

\(\dfrac{1}{\sqrt{a^2 + x^2}}\)

\(\log_\epsilon (x + \sqrt{a^2 + x^2}) + C\)

\(\mp \dfrac{b}{(a \pm bx)^2}\)

\(\dfrac{1}{a \pm bx}\)

\(\pm \dfrac{1}{b} \log_\epsilon (a \pm bx) + C\)

\(-\dfrac{3a^2x}{(a^2 + x^2)^{\frac{5}{2}}}\)

\(\dfrac{a^2}{(a^2 + x^2)^{\frac{3}{2}}}\)

\(\dfrac{x}{\sqrt{a^2 + x^2}} + C\)

\(a \cdot \cos ax\)

\(\sin ax\)

\(-\dfrac{1}{a} \cos ax + C\)

\(-a \cdot \sin ax\)

\(\cos ax\)

\(\dfrac{1}{a} \sin ax + C\)

\(a \cdot \sec^2ax\)

\(\tan ax\)

\(-\dfrac{1}{a} \log_\epsilon \cos ax + C\)

\(\sin 2x\)

\(\sin^2 x\)

\(\dfrac{x}{2} – \dfrac{\sin 2x}{4} + C\)

\(-\sin 2x\)

\(\cos^2 x\)

$\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C $

$ 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 $

 

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