What is the integral of sin(4x^2) ?

The integral of sin(4x^2) is (sqrt(pi))/(2sqrt(2))\S(2sqrt(2/pi)x)+C

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integral of sin(4x^2)

\int sin (4 x^{2}) d x

\frac{π}{2 2} § (2 \frac{2}{π} x) + C

Solution steps

\int sin (4 x^{2}) d x

Apply Integral Substitution

= \int \frac{1}{2} sin (u^{2}) d u

Take the constant out:

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

= \frac{1}{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)

= \frac{1}{2} \frac{π}{2} § (\frac{2}{π} u)

Substitute back

u = 2 x

= \frac{1}{2} \frac{π}{2} § (\frac{2}{π} \cdot 2 x)

Simplify

\frac{1}{2} \frac{π}{2} § (\frac{2}{π} \cdot 2 x) : \frac{π}{2 2} § (2 \frac{2}{π} x)

= \frac{π}{2 2} § (2 \frac{2}{π} x)

Add a constant to the solution

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

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What is the integral of sin(4x^2) ?
The integral of sin(4x^2) is (sqrt(pi))/(2sqrt(2))\S(2sqrt(2/pi)x)+C