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

The integral of (sin^2(x))/x is 1/2 (ln|x|-\Ci(2x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{1}{2} (ln ∣ x ∣ - Ci (2 x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 - cos ( 2 x )}{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{1 - cos ( 2 x )}{x} d x

Expand

\frac{1 - cos ( 2 x )}{x} : \frac{1}{x} - \frac{cos ( 2 x )}{x}

= \frac{1}{2} \cdot \int \frac{1}{x} - \frac{cos ( 2 x )}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{1}{x} d x - \int \frac{cos ( 2 x )}{x} d x)

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{cos ( 2 x )}{x} d x = Ci (2 x)

= \frac{1}{2} (ln ∣ x ∣ - Ci (2 x))

Add a constant to the solution

= \frac{1}{2} (ln ∣ x ∣ - Ci (2 x)) + C

\int e^{- 2 y} d y

\frac{d}{d x} (900 x - \frac{1}{3} x^{3} + 2500)

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

5 y^{^{''}} + 5 y^{^{'}} - 10 y = 0

\int_{- \infty}^{0} (4 - 3 x) e^{x} d x

What is the integral of (sin^2(x))/x ?
The integral of (sin^2(x))/x is 1/2 (ln|x|-\Ci(2x))+C