What is the integral of e^{-1/2 t^2} ?

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

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integral of e^{-1/2 t^2}

\int e^{- \frac{1}{2} t^{2}} d t

\frac{π}{2} erf (\frac{t}{2}) + C

Solution steps

\int e^{- \frac{1}{2} t^{2}} d t

Simplify

- \frac{1}{2} t^{2} : - \frac{t ^{2}}{2}

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

Apply exponent rule:

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

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

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

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} erf (u)

= 2 \frac{π}{2} erf (u)

Substitute back

u = \frac{t}{2}

= 2 \frac{π}{2} erf (\frac{t}{2})

Simplify

2 \frac{π}{2} erf (\frac{t}{2}) : \frac{π}{2} erf (\frac{t}{2})

= \frac{π}{2} erf (\frac{t}{2})

Add a constant to the solution

= \frac{π}{2} erf (\frac{t}{2}) + C

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