What is the integral of (ln(x))/(xsqrt(16-ln(x))) ?

The integral of (ln(x))/(xsqrt(16-ln(x))) is 2/3 (16-ln(x))^{3/2}-32sqrt(16-ln(x))+C

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integral of (ln(x))/(xsqrt(16-ln(x)))

\int \frac{ln ( x )}{x 16 - ln ( x )} d x

\frac{2}{3} (16 - ln (x))^{\frac{3}{2}} - 32 16 - ln (x) + C

Solution steps

\int \frac{ln ( x )}{x 16 - ln ( x )} d x

Apply u-substitution

: \int - \frac{- u + 16}{u} d u

= \int - \frac{- u + 16}{u} d u

Take the constant out:

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

= - \int \frac{- u + 16}{u} d u

Expand

\frac{- u + 16}{u} : - u + \frac{16}{u}

= - \int - u + \frac{16}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \int u d u + \int \frac{16}{u} d u)

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

\int \frac{16}{u} d u = 32 u

= - (- \frac{2}{3} u^{\frac{3}{2}} + 32 u)

Substitute back

u = 16 - ln (x)

= - (- \frac{2}{3} (16 - ln (x))^{\frac{3}{2}} + 32 16 - ln (x))

Simplify

= \frac{2}{3} (16 - ln (x))^{\frac{3}{2}} - 32 16 - ln (x)

Add a constant to the solution

= \frac{2}{3} (16 - ln (x))^{\frac{3}{2}} - 32 16 - ln (x) + C

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What is the integral of (ln(x))/(xsqrt(16-ln(x))) ?
The integral of (ln(x))/(xsqrt(16-ln(x))) is 2/3 (16-ln(x))^{3/2}-32sqrt(16-ln(x))+C