What is the integral of 1/(x^3sqrt(ln(x^5))) ?

The integral of 1/(x^3sqrt(ln(x^5))) is (sqrt(pi))/(sqrt(10))\erf(sqrt(2)sqrt(ln(x)))+C

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integral of 1/(x^3sqrt(ln(x^5)))

\int \frac{1}{x ^{3} ln ( x ^{5} )} d x

\frac{π}{10} erf (2 ln (x)) + C

Solution steps

\int \frac{1}{x ^{3} ln ( x ^{5} )} d x

ln (x^{5}) = 5 ln (x)

= \int \frac{1}{5 x ^{3} ln ( x )} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{x ^{3} ln ( x )} d x

Apply u-substitution

: \int \frac{2}{e ^{2 u^{2}}} d u

= \frac{1}{5} \cdot \int \frac{2}{e ^{2 u^{2}}} d u

Take the constant out:

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

= \frac{1}{5} \cdot 2 \cdot \int \frac{1}{e ^{2 u^{2}}} d u

Apply Integration By Parts

: \frac{u}{e ^{2 u^{2}}} - \int - 4 e^{- 2 u^{2}} u^{2} d u

= \frac{1}{5} \cdot 2 (\frac{u}{e ^{2 u^{2}}} - \int - 4 e^{- 2 u^{2}} u^{2} d u)

\int - 4 e^{- 2 u^{2}} u^{2} d u = e^{- 2 u^{2}} u - \frac{π}{2 2} erf (2 u)

= \frac{1}{5} \cdot 2 (\frac{u}{e ^{2 u^{2}}} - (e^{- 2 u^{2}} u - \frac{π}{2 2} erf (2 u)))

Substitute back

u = ln (x)

= \frac{1}{5} \cdot 2 (\frac{ln ( x )}{e ^{2 (l n (x))^{2}}} - (e^{- 2 (l n (x))^{2}} ln (x) - \frac{π}{2 2} erf (2 ln (x))))

Simplify

\frac{1}{5} \cdot 2 (\frac{ln ( x )}{e ^{2 (l n (x))^{2}}} - (e^{- 2 (l n (x))^{2}} ln (x) - \frac{π}{2 2} erf (2 ln (x)))) : \frac{π}{10} erf (2 ln (x))

= \frac{π}{10} erf (2 ln (x))

Add a constant to the solution

= \frac{π}{10} erf (2 ln (x)) + C

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What is the integral of 1/(x^3sqrt(ln(x^5))) ?
The integral of 1/(x^3sqrt(ln(x^5))) is (sqrt(pi))/(sqrt(10))\erf(sqrt(2)sqrt(ln(x)))+C