What is the integral of sin((x^2)/2) ?

The integral of sin((x^2)/2) is sqrt(pi)\S(sqrt(2/pi) x/(sqrt(2)))+C

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integral of sin((x^2)/2)

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

π § (\frac{2}{π} \frac{x}{2}) + C

Solution steps

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

Apply u-substitution

: \int sin (u^{2}) 2 d u

= \int sin (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 sin (u^{2}) d u

Use the nonelementary integral:

\int sin (x^{2}) d x = \frac{π}{2} § (\frac{2}{π} x)

= 2 \frac{π}{2} § (\frac{2}{π} u)

Substitute back

u = \frac{x}{2}

= 2 \frac{π}{2} § (\frac{2}{π} \frac{x}{2})

Simplify

2 \frac{π}{2} § (\frac{2}{π} \frac{x}{2}) : π § (\frac{2}{π} \frac{x}{2})

= π § (\frac{2}{π} \frac{x}{2})

Add a constant to the solution

= π § (\frac{2}{π} \frac{x}{2}) + C

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What is the integral of sin((x^2)/2) ?
The integral of sin((x^2)/2) is sqrt(pi)\S(sqrt(2/pi) x/(sqrt(2)))+C