What is the integral of (sqrt(ln(x)+5))/x ?

The integral of (sqrt(ln(x)+5))/x is 2/3 (ln(x)+5)^{3/2}+C

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integral of (sqrt(ln(x)+5))/x

\int \frac{ln ( x ) + 5}{x} d x

\frac{2}{3} (ln (x) + 5)^{\frac{3}{2}} + C

Solution steps

\int \frac{ln ( x ) + 5}{x} d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = ln (x) + 5

= \frac{2}{3} (ln (x) + 5)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (ln (x) + 5)^{\frac{3}{2}} + C

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What is the integral of (sqrt(ln(x)+5))/x ?
The integral of (sqrt(ln(x)+5))/x is 2/3 (ln(x)+5)^{3/2}+C