$$\dfrac{dy}{dx}$$  ⬅️⬅️⬅️⬅️⬅️ $$y$$ ➡️➡️➡️➡️➡️➡️  $$\displaystyle \int y\, dx$$

#### Algebraic.

$$1$$ $$x$$ $$\frac{1}{2} x^2 + C$$
$$0$$ $$a$$ $$ax + C$$
$$1$$ $$x \pm a$$ $$\frac{1}{2} x^2 \pm ax + C$$
$$a$$ $$ax$$ $$\frac{1}{2} ax^2 + C$$
$$2x$$ $$x^2$$ $$\frac{1}{3} x^3 + C$$
$$nx^{n-1}$$ $$x^n$$ $$\dfrac{1}{n+1} x^{n+1} + C$$
$$-x^{-2}$$ $$x^{-1}$$ $$\ln x + C$$
$$\dfrac{du}{dx} \pm \dfrac{dv}{dx} \pm \dfrac{dw}{dx}$$ $$u \pm v \pm w$$ $$\displaystyle \int u\, dx \pm \int v\, dx \pm \int w\, dx$$
$$u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}$$ $$uv$$ 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.

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

#### Trigonometrical.

$$\cos x$$ $$\sin x$$ $$-\cos x + C$$
$$-\sin x$$ $$\cos x$$ $$\sin x + C$$
$$\sec^2 x$$ $$\tan x$$ $$-\ln \cos x + C$$

#### Circular (Inverse).

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

#### Hyperbolic.

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

#### Miscellaneous.

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

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

$\ln (x+a) + C$

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

$$\dfrac{1}{\sqrt{a^2 + x^2}}$$

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

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

$$\dfrac{1}{a \pm bx}$$

$$\pm \dfrac{1}{b} \ln (a \pm bx) + C$$

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

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

$$\dfrac{x}{\sqrt{a^2 + x^2}} + C$$

$$a \cdot \cos ax$$

$$\sin ax$$

$$-\dfrac{1}{a} \cos ax + C$$

$$-a \cdot \sin ax$$

$$\cos ax$$

$$\dfrac{1}{a} \sin ax + C$$

$$a \cdot \sec^2ax$$

$$\tan ax$$

$$-\dfrac{1}{a} \ln \cos ax + C$$

$$\sin 2x$$

$$\sin^2 x$$

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

$$-\sin 2x$$

$$\cos^2 x$$

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

$2a\cdot\sin 2ax$

$\sin^2 ax$

$\dfrac{x}{2} – \dfrac{\sin 2ax}{4a} + C$

$-2a\cdot\sin 2ax$

$\cos^2 ax$

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