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

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

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

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

arctan (ln (x)) + C

Solution steps

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

Simplify

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= arctan (u)

Substitute back

u = ln (x)

= arctan (ln (x))

Add a constant to the solution

= arctan (ln (x)) + C

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