What is the integral of (ln(x))/(x(1+ln^2(x))) ?

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

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integral of (ln(x))/(x(1+ln^2(x)))

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

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

Solution steps

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

Apply u-substitution

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

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

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}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + ln^{2} (x)

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

Add a constant to the solution

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

\int_{0}^{15} 1000 e^{- 0.05 x} d x

\frac{d}{d x} ((5 x^{2} - 2) (x^{3} + 3 x))

\frac{d}{d x} (4 x^{3} + 2 + \frac{1}{x})

\frac{d y}{d x} = 2 x e^{5 y}

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