Peterson Ward
01/03/2024 · Primary School
a) \( \int\left(\frac{5}{\sqrt[1]{s^{2}}}+\frac{2}{\sqrt{s^{3}}}\right) d s \)
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Step-by-step Solution
Para resolver la integral
\[
\int\left(\frac{5}{\sqrt[1]{s^{2}}}+\frac{2}{\sqrt{s^{3}}}\right) d s,
\]
primero simplificamos las expresiones dentro de la integral. Notamos que \(\sqrt[1]{s^{2}} = s^{2}\) y \(\sqrt{s^{3}} = s^{3/2}\). Por lo tanto, podemos reescribir la integral como:
\[
\int\left(\frac{5}{s^{2}}+\frac{2}{s^{3/2}}\right) d s.
\]
Ahora, podemos separar la integral en dos partes:
\[
\int \frac{5}{s^{2}} d s + \int \frac{2}{s^{3/2}} d s.
\]
Calculamos cada integral por separado.
1. Para la primera integral:
\[
\int \frac{5}{s^{2}} d s = 5 \int s^{-2} d s = 5 \left(-s^{-1}\right) = -\frac{5}{s}.
\]
2. Para la segunda integral:
\[
\int \frac{2}{s^{3/2}} d s = 2 \int s^{-3/2} d s = 2 \left(-\frac{2}{\sqrt{s}}\right) = -\frac{4}{\sqrt{s}}.
\]
Ahora, sumamos los resultados de ambas integrales:
\[
-\frac{5}{s} - \frac{4}{\sqrt{s}} + C,
\]
donde \(C\) es la constante de integración.
Por lo tanto, la solución de la integral es:
\[
-\frac{5}{s} - \frac{4}{\sqrt{s}} + C.
\]
Quick Answer
La solución de la integral es \(-\frac{5}{s} - \frac{4}{\sqrt{s}} + C\).
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