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

The integral of 1/(x^5sqrt(ln(x))) is (sqrt(pi))/2 \erf(sqrt(4)sqrt(ln(x)))+C

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

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

\frac{π}{2} erf (4 ln (x)) + C

Solution steps

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

Apply u-substitution

= \int \frac{2}{e ^{4 u^{2}}} d u

Take the constant out:

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

= 2 \cdot \int \frac{1}{e ^{4 u^{2}}} d u

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{e ^{4 u^{2}}} = e^{- 4 u^{2}}

= 2 \cdot \int e^{- 4 u^{2}} d u

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

Apply u-substitution

= 2 \cdot \int e^{- v^{2}} \frac{1}{2} d v

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int e^{- v^{2}} d v

Use the nonelementary integral:

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

= 2 \cdot \frac{1}{2} \cdot \frac{π}{2} erf (v)

Substitute back

= 2 \cdot \frac{1}{2} \cdot \frac{π}{2} erf (4 ln (x))

Simplify

2 \cdot \frac{1}{2} \cdot \frac{π}{2} erf (4 ln (x)) : \frac{π}{2} erf (4 ln (x))

= \frac{π}{2} erf (4 ln (x))

Add a constant to the solution

= \frac{π}{2} erf (4 ln (x)) + C

What is the integral of 1/(x^5sqrt(ln(x))) ?
The integral of 1/(x^5sqrt(ln(x))) is (sqrt(pi))/2 \erf(sqrt(4)sqrt(ln(x)))+C