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

The integral of (sin(x^2))/(2x) is 1/4 \Si(x^2)+C

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

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

\frac{1}{4} Si (x^{2}) + C

Solution steps

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

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 ( x ^{2} )}{x} d x

Apply u-substitution

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

Use the nonelementary integral:

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

= \frac{1}{2} \cdot \frac{1}{2} Si (u)

Substitute back

u = x^{2}

= \frac{1}{2} \cdot \frac{1}{2} Si (x^{2})

Simplify

\frac{1}{2} \cdot \frac{1}{2} Si (x^{2}) : \frac{1}{4} Si (x^{2})

= \frac{1}{4} Si (x^{2})

Add a constant to the solution

= \frac{1}{4} Si (x^{2}) + C

What is the integral of (sin(x^2))/(2x) ?
The integral of (sin(x^2))/(2x) is 1/4 \Si(x^2)+C