What is the integral of-(2ln(x))/x ?

The integral of-(2ln(x))/x is -ln^2(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of-(2ln(x))/x

\int - \frac{2 ln ( x )}{x} d x

- ln^{2} (x) + C

Solution steps

\int - \frac{2 ln ( x )}{x} d x

Take the constant out:

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

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

Apply u-substitution

: \int u d u

= - 2 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - 2 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= - 2 \cdot \frac{ln ^{2} ( x )}{2}

Simplify

- 2 \cdot \frac{ln ^{2} ( x )}{2} : - ln^{2} (x)

= - ln^{2} (x)

Add a constant to the solution

= - ln^{2} (x) + C

\int tan^{3} (2 x) sec (2 x) d x

\frac{d}{d x} ((1 + \frac{1}{x})^{x})

\frac{d y}{d x} + (3 x y^{3}) = 0, y (0) = 1

\frac{d}{d x} (\frac{3}{6 x + 3})

\int_{0}^{5} \frac{1}{x ^{\frac{21}{22}}} d x