Solve operatorname { erf } x = 2 / sqrt { pi } ∫ (from 0 to x) of e ^ - t ^ 2 dt

$$\operatorname { erf } x = \frac { 2 } { \sqrt { \pi } } \int _ { 0 } ^ { x } e ^ { - t ^ { 2 } } d t$$

Solve for f

$$\left\{\begin{matrix}f=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{e\sqrt{\pi }rx}\text{, }&x\neq 0\text{ and }r\neq 0\\f\in \mathrm{R}\text{, }&\left(\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t=0\text{ and }r=0\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Express \frac{2}{\sqrt{\pi }}\int _{0}^{x}e^{-t^{2}}\mathrm{d}t as a single fraction.

$$erfx=\frac{2\int _{0}^{x}e^{-t^{2}}\mathrm{d}t}{\sqrt{\pi }}$$

The equation is in standard form.

$$erxf=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }}$$

Divide both sides by erx.

$$\frac{erxf}{erx}=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }erx}$$

Dividing by erx undoes the multiplication by erx.

$$f=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }erx}$$

Divide \frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }} by erx.

$$f=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{e\sqrt{\pi }rx}$$

Solve for r

$$\left\{\begin{matrix}r=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{e\sqrt{\pi }fx}\text{, }&x\neq 0\text{ and }f\neq 0\\r\in \mathrm{R}\text{, }&\left(\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t=0\text{ and }f=0\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Express \frac{2}{\sqrt{\pi }}\int _{0}^{x}e^{-t^{2}}\mathrm{d}t as a single fraction.

$$erfx=\frac{2\int _{0}^{x}e^{-t^{2}}\mathrm{d}t}{\sqrt{\pi }}$$

The equation is in standard form.

$$efxr=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }}$$

Divide both sides by efx.

$$\frac{efxr}{efx}=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }efx}$$

Dividing by efx undoes the multiplication by efx.

$$r=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }efx}$$

Divide \frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{\sqrt{\pi }} by efx.

$$r=\frac{2\int _{0}^{x}\frac{1}{e^{t^{2}}}\mathrm{d}t}{e\sqrt{\pi }fx}$$