What is the integral of ln(x)sqrt(x^{15)} ?

The integral of ln(x)sqrt(x^{15)} is 2/17 x^{17/2}ln(x)-4/289 x^{17/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ln(x)sqrt(x^{15)}

\int ln (x) x^{15} d x

\frac{2}{17} x^{\frac{17}{2}} ln (x) - \frac{4}{289} x^{\frac{17}{2}} + C

Solution steps

\int ln (x) x^{15} d x

Apply u-substitution

= \int e^{\frac{17 u}{2}} u d u

Apply Integration By Parts

= \frac{2}{17} e^{\frac{17 u}{2}} u - \int \frac{2}{17} e^{\frac{17 u}{2}} d u

\int \frac{2}{17} e^{\frac{17 u}{2}} d u = \frac{4}{289} e^{\frac{17 u}{2}}

= \frac{2}{17} e^{\frac{17 u}{2}} u - \frac{4}{289} e^{\frac{17 u}{2}}

Substitute back

u = ln (x)

= \frac{2}{17} e^{\frac{17 l n ( x )}{2}} ln (x) - \frac{4}{289} e^{\frac{17 l n ( x )}{2}}

Simplify

\frac{2}{17} e^{\frac{17 l n ( x )}{2}} ln (x) - \frac{4}{289} e^{\frac{17 l n ( x )}{2}} : \frac{2}{17} x^{\frac{17}{2}} ln (x) - \frac{4}{289} x^{\frac{17}{2}}

= \frac{2}{17} x^{\frac{17}{2}} ln (x) - \frac{4}{289} x^{\frac{17}{2}}

Add a constant to the solution

= \frac{2}{17} x^{\frac{17}{2}} ln (x) - \frac{4}{289} x^{\frac{17}{2}} + C

What is the integral of ln(x)sqrt(x^{15)} ?
The integral of ln(x)sqrt(x^{15)} is 2/17 x^{17/2}ln(x)-4/289 x^{17/2}+C