Pregunta
\( \int \sqrt[ 5 ] { r ^ { 4 } } d r = \square \)
Ask by Davison Love. in the United States
Jan 30,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
The integral of \( \sqrt[5]{r^4} \) with respect to \( r \) is \( \frac{5}{9} r^{\frac{9}{5}} + C \).
Solución
To evaluate the integral:
\[
\int \sqrt[5]{r^4} \, dr
\]
we can first rewrite the fifth root in exponent form:
\[
\sqrt[5]{r^4} = r^{4/5}
\]
Now, the integral becomes:
\[
\int r^{4/5} \, dr
\]
Using the power rule for integration, which states:
\[
\int r^n \, dr = \frac{r^{n+1}}{n+1} + C \quad \text{(where } n \neq -1 \text{)}
\]
Here, \( n = \frac{4}{5} \). Applying the rule:
\[
\frac{r^{\frac{4}{5} + 1}}{\frac{4}{5} + 1} + C = \frac{r^{\frac{9}{5}}}{\frac{9}{5}} + C = \frac{5}{9} r^{\frac{9}{5}} + C
\]
**Final Answer:**
\[
\int \sqrt[5]{r^4} \, dr = \frac{5}{9}\, r^{\,9/5} + C
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Extra Insights
To tackle the integral \( \int \sqrt[5]{r^4} \, dr \), we can simplify the expression inside the integral. Recall that \( \sqrt[5]{r^4} = r^{\frac{4}{5}} \). So, we can rewrite the integral as \( \int r^{\frac{4}{5}} \, dr \). Now, applying the power rule for integrals, we have: \[ \int r^{\frac{4}{5}} \, dr = \frac{r^{\frac{4}{5} + 1}}{\frac{4}{5} + 1} + C = \frac{r^{\frac{9}{5}}}{\frac{9}{5}} + C = \frac{5}{9} r^{\frac{9}{5}} + C \] And there you have it! This is the result of your integral: \[ \int \sqrt[5]{r^{4}} \, dr = \frac{5}{9} r^{\frac{9}{5}} + C \] Just don't forget the constant of integration \( C \)!
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