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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

= \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

\int \frac{\frac{1}{1 - x ^{2}}}{x ^{2}} d x

\int 1 - \frac{x ^{2}}{a ^{2}} d x

\int \frac{x ^{2} + 5}{2 x} d x

\int \frac{x ^{3} + x - 1}{( x ^{2} + 1 ) ^{2}} d x

\int \frac{1}{- 4 x} d x

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