What is the integral of e^{-((x-3)^2)/2} ?

The integral of e^{-((x-3)^2)/2} is (pi/2)^{1/2}\erf(1/(sqrt(2))(x-3))+C

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integral of e^{-((x-3)^2)/2}

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

(\frac{π}{2})^{\frac{1}{2}} erf (\frac{1}{2} (x - 3)) + C

Solution steps

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

x - 3 = 3 (\frac{x}{3} - 1)

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

Complete the square

- \frac{( 3 ( \frac{x}{3} - 1 ) ) ^{2}}{2} : - \frac{1}{2} (x - 3)^{2}

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

Apply u-substitution

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

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

Apply exponent rule:

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

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

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

Apply u-substitution

: \int e^{- v^{2}} 2 d v

= \int e^{- v^{2}} 2 d v

Take the constant out:

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

= 2 \cdot \int e^{- v^{2}} d v

Use the nonelementary integral:

\int e^{- v^{2}} d v = \frac{π}{2} erf (v)

= 2 \frac{π}{2} erf (v)

Substitute back

= 2 \frac{π}{2} erf (\frac{x - 3}{2})

Simplify

2 \frac{π}{2} erf (\frac{x - 3}{2}) : (\frac{π}{2})^{\frac{1}{2}} erf (\frac{1}{2} (x - 3))

= (\frac{π}{2})^{\frac{1}{2}} erf (\frac{1}{2} (x - 3))

Add a constant to the solution

= (\frac{π}{2})^{\frac{1}{2}} erf (\frac{1}{2} (x - 3)) + C

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What is the integral of e^{-((x-3)^2)/2} ?
The integral of e^{-((x-3)^2)/2} is (pi/2)^{1/2}\erf(1/(sqrt(2))(x-3))+C