What is the integral of (ln(x))/(xsqrt(1+7ln^2(x))) ?

The integral of (ln(x))/(xsqrt(1+7ln^2(x))) is 1/7 sqrt(1+7ln^2(x))+C

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integral of (ln(x))/(xsqrt(1+7ln^2(x)))

\int \frac{ln ( x )}{x 1 + 7 ln ^{2} ( x )} d x

\frac{1}{7} 1 + 7 ln^{2} (x) + C

Solution steps

\int \frac{ln ( x )}{x 1 + 7 ln ^{2} ( x )} d x

Apply u-substitution

= \int \frac{u}{1 + 7 u ^{2}} d u

Apply u-substitution

= \int \frac{1}{14 v} d v

Take the constant out:

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

= \frac{1}{14} \cdot \int \frac{1}{v} d v

Apply the Power Rule

= \frac{1}{14} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{14} \cdot 2 (1 + 7 ln^{2} (x))^{\frac{1}{2}}

Simplify

\frac{1}{14} \cdot 2 (1 + 7 ln^{2} (x))^{\frac{1}{2}} : \frac{1}{7} 1 + 7 ln^{2} (x)

= \frac{1}{7} 1 + 7 ln^{2} (x)

Add a constant to the solution

= \frac{1}{7} 1 + 7 ln^{2} (x) + C

What is the integral of (ln(x))/(xsqrt(1+7ln^2(x))) ?
The integral of (ln(x))/(xsqrt(1+7ln^2(x))) is 1/7 sqrt(1+7ln^2(x))+C