What is the integral of (2+ln(7x))/(3x) ?

The integral of (2+ln(7x))/(3x) is 1/3 (2ln|x|+(ln^2(7x))/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2+ln(7x))/(3x)

\int \frac{2 + ln ( 7 x )}{3 x} d x

\frac{1}{3} (2 ln ∣ x ∣ + \frac{ln ^{2} ( 7 x )}{2}) + C

Solution steps

\int \frac{2 + ln ( 7 x )}{3 x} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{2 + ln ( 7 x )}{x} d x

Expand

\frac{2 + ln ( 7 x )}{x} : \frac{2}{x} + \frac{ln ( 7 x )}{x}

= \frac{1}{3} \cdot \int \frac{2}{x} + \frac{ln ( 7 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}{3} (\int \frac{2}{x} d x + \int \frac{ln ( 7 x )}{x} d x)

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{ln ( 7 x )}{x} d x = \frac{ln ^{2} ( 7 x )}{2}

= \frac{1}{3} (2 ln ∣ x ∣ + \frac{ln ^{2} ( 7 x )}{2})

Add a constant to the solution

= \frac{1}{3} (2 ln ∣ x ∣ + \frac{ln ^{2} ( 7 x )}{2}) + C

y^{^{'}} + 4 x^{3} y = x^{3}

\frac{\partial}{\partial x} (- 5 x y + 2)

\frac{d}{d x} (4 x^{2} - 1)

\int \frac{1}{( 3 n + 5 ) ^{4}}

\int \frac{x ^{2}}{( x ^{2} + 7 ) ^{2}} d x

What is the integral of (2+ln(7x))/(3x) ?
The integral of (2+ln(7x))/(3x) is 1/3 (2ln|x|+(ln^2(7x))/2)+C