use integration by parts to evaluate the integral:\n$$\\int \\ln(r) \\sqrt{r^{19}} dr$$
use integration by parts to evaluate the integral:\n$$\\int \\ln(r) \\sqrt{r^{19}} dr$$
Answer
Explanation:
Step1: Rewrite the integrand using exponents
$$ \int \ln(r) \cdot r^{19/2} , dr $$
Step2: Identify $u$ and $dv$ for integration by parts
Let $u = \ln(r)$ and $dv = r^{19/2} , dr$.
Step3: Differentiate $u$ and integrate $dv$
$$ du = \frac{1}{r} , dr, \quad v = \int r^{19/2} , dr = \frac{r^{21/2}}{21/2} = \frac{2}{21}r^{21/2} $$
Step4: Apply the integration by parts formula
$$ \int u , dv = uv - \int v , du $$ $$ \frac{2}{21}r^{21/2} \ln(r) - \int \frac{2}{21}r^{21/2} \cdot \frac{1}{r} , dr $$
Step5: Simplify the remaining integral
$$ \frac{2}{21}r^{21/2} \ln(r) - \frac{2}{21} \int r^{19/2} , dr $$
Step6: Evaluate the final integral and add constant
$$ \frac{2}{21}r^{21/2} \ln(r) - \frac{2}{21} \left( \frac{2}{21}r^{21/2} \right) + C $$
Step7: Simplify the expression
$$ \frac{2}{21}r^{21/2} \ln(r) - \frac{4}{441}r^{21/2} + C $$
Answer:
\frac{2}{21}r^{21/2} \ln(r) - \frac{4}{441}r^{21/2} + C