What is the integral of (ln(x))/(x(4+ln^2(x))) ?

The integral of (ln(x))/(x(4+ln^2(x))) is 1/2 ln|4+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 (ln(x))/(x(4+ln^2(x)))

\int \frac{ln ( x )}{x ( 4 + ln ^{2} ( x ) )} d x

\frac{1}{2} ln 4 + ln^{2} (x) + C

Solution steps

\int \frac{ln ( x )}{x ( 4 + ln ^{2} ( x ) )} d x

Apply u-substitution

: \int \frac{u}{4 + u ^{2}} d u

= \int \frac{u}{4 + u ^{2}} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= \int \frac{1}{2 v} d v

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}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} ln ∣ v ∣

Substitute back

= \frac{1}{2} ln 4 + ln^{2} (x)

Add a constant to the solution

= \frac{1}{2} ln 4 + ln^{2} (x) + C

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

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

\int 30 sin (4 x - 2) cos (x + 1) d x

\int e^{(x^{3} - 6 x)} (x^{2} - 2) d x

\int \frac{x - 4}{( x + 5 ) ^{2}} d x

What is the integral of (ln(x))/(x(4+ln^2(x))) ?
The integral of (ln(x))/(x(4+ln^2(x))) is 1/2 ln|4+ln^2(x)|+C