What is the integral of e^{((x^2))/2} ?

The integral of e^{((x^2))/2} is (pi/2)^{1/2}\erfi(x/(sqrt(2)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of e^{((x^2))/2}

\int e^{\frac{( x ^{2} )}{2}} d x

(\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2}) + C

Solution steps

\int e^{\frac{x ^{2}}{2}} d x

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{2} = (\frac{x}{2})^{2}

= \int e^{(\frac{x}{2})^{2}} d x

Apply u-substitution

: \int e^{u^{2}} 2 d u

= \int e^{u^{2}} 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int e^{u^{2}} d u

Use the nonelementary integral:

\int e^{u^{2}} d u = \frac{π}{2} erfi (u)

= 2 \frac{π}{2} erfi (u)

Substitute back

u = \frac{x}{2}

= 2 \frac{π}{2} erfi (\frac{x}{2})

Simplify

2 \frac{π}{2} erfi (\frac{x}{2}) : (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2})

= (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2})

Add a constant to the solution

= (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2}) + C

\int \frac{20 x ^{4}}{2 x ^{5} ln ( 3 )} d x

\int sin^{6} (\frac{x}{4}) cos (\frac{x}{4}) d x

\int \frac{1}{( x + 1 ) ^{2} ( x ^{2} + 2 ) ( x - 3 )} d x

\int (x^{2} - 2 x + 3) cos (2 x) d x

\int \frac{( x ^{2} - 1 ) ^{2}}{x} d x

What is the integral of e^{((x^2))/2} ?
The integral of e^{((x^2))/2} is (pi/2)^{1/2}\erfi(x/(sqrt(2)))+C