What is the integral of xsqrt(ln(x)) ?

The integral of xsqrt(ln(x)) is 1/(4sqrt(2))(2sqrt(2)x^2sqrt(ln(x))-sqrt(pi)\erfi(sqrt(2ln(x))))+C

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integral of xsqrt(ln(x))

\int x ln (x) d x

\frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x))) + C

Solution steps

\int x ln (x) d x

Apply u-substitution

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

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

Apply u-substitution

: \int e^{v} \frac{v}{2} \frac{1}{2} d v

= \int e^{v} \frac{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

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

Apply Integration By Parts

: e^{v} \frac{v}{2} - \int \frac{e ^{v}}{2 2 v} d v

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

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

= \frac{1}{2} (e^{v} \frac{v}{2} - \frac{π}{2 2} erfi (v))

Substitute back

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

Simplify

\frac{1}{2} (e^{2 l n (x)} \frac{2 ln ( x )}{2} - \frac{π}{2 2} erfi (2 ln (x))) : \frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x)))

= \frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x)))

Add a constant to the solution

= \frac{1}{4 2} (22 x^{2} ln (x) - π erfi (2 ln (x))) + C

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