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

The integral of (sin(pix^2))/x is 1/2 \Si(pix^2)+C

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

\int \frac{sin ( π x ^{2} )}{x} d x

\frac{1}{2} Si (π x^{2}) + C

Solution steps

\int \frac{sin ( π x ^{2} )}{x} d x

Apply u-substitution

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

Apply u-substitution

= \frac{1}{2} \cdot \int \frac{sin ( v )}{v} d v

Use the nonelementary integral:

\int \frac{sin ( x )}{x} d x = Si (x)

= \frac{1}{2} Si (v)

Substitute back

= \frac{1}{2} Si (π x^{2})

Add a constant to the solution

= \frac{1}{2} Si (π x^{2}) + C

What is the integral of (sin(pix^2))/x ?
The integral of (sin(pix^2))/x is 1/2 \Si(pix^2)+C