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

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

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

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

4 arctan (ln (x)) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 4 \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

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

= 4 arctan (u)

Substitute back

u = ln (x)

= 4 arctan (ln (x))

Add a constant to the solution

= 4 arctan (ln (x)) + C

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