What is the integral of 1/(x(4+(3ln(x))^2)) ?

The integral of 1/(x(4+(3ln(x))^2)) is 1/6 arctan(3/2 ln(x))+C

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integral of 1/(x(4+(3ln(x))^2))

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

\frac{1}{6} arctan (\frac{3}{2} ln (x)) + C

Solution steps

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

Simplify

\frac{1}{x ( 4 + ( 3 ln ( x ) ) ^{2} )} : \frac{1}{x ( 4 + 9 ln ^{2} ( x ) )}

= \int \frac{1}{x ( 4 + 9 ln ^{2} ( x ) )} d x

Apply u-substitution

: \int \frac{1}{4 + 9 u ^{2}} d u

= \int \frac{1}{4 + 9 u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{6 ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

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

= \frac{1}{6} arctan (v)

Substitute back

= \frac{1}{6} arctan (\frac{3}{2} ln (x))

Add a constant to the solution

= \frac{1}{6} arctan (\frac{3}{2} ln (x)) + C

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What is the integral of 1/(x(4+(3ln(x))^2)) ?
The integral of 1/(x(4+(3ln(x))^2)) is 1/6 arctan(3/2 ln(x))+C