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

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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

= \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

= \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

\frac{3}{( 4 x + 1 ) ^{2}}

\frac{d}{d x} (\frac{x ^{3}}{3} + 1)

t \to 0 lim (\frac{t}{t ^{2} - t})

\frac{d}{d x} (x + 1 - e^{x})

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