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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Integral Substitution

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

Use the common integral:

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

= \frac{1}{2} arctan (v)

Substitute back

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

Add a constant to the solution

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

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