What is the integral from 0 to 1 of (4sin(x^2))/x ?

The integral from 0 to 1 of (4sin(x^2))/x is 1.89216…

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from 0 to 1 of (4sin(x^2))/x

\int_{0}^{1} \frac{4 sin ( x ^{2} )}{x} d x

1.89216 \dots

Solution steps

\int_{0}^{1} \frac{4 sin ( x ^{2} )}{x} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{1} \frac{sin ( x ^{2} )}{x} d x

Apply u-substitution

= 4 \cdot \int_{0}^{1} \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

= 4 \cdot \frac{1}{2} \cdot \int_{0}^{1} \frac{sin ( u )}{u} d u

Use the nonelementary integral:

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

= 4 \cdot \frac{1}{2} [Si (u)]_{0}^{1}

Simplify

4 \cdot \frac{1}{2} [Si (u)]_{0}^{1} : 2 [Si (u)]_{0}^{1}

= 2 [Si (u)]_{0}^{1}

Compute the boundaries:

0.9460830704

= 2 \cdot 0.9460830704

Simplify

= 1.89216 \dots

What is the integral from 0 to 1 of (4sin(x^2))/x ?
The integral from 0 to 1 of (4sin(x^2))/x is 1.89216…