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

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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

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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