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

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

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

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

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

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

Apply the Power Rule

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

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

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

Simplify

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

= 1 + ln^{2} (x)

Add a constant to the solution

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

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