Perkins Guerrero
07/31/2024 · Elementary School
Hallar la integral por sustitución de las Funciones a) \( F(x)=\left(6 x^{5}+5 x^{4}\right) e^{\left(x^{6}+x^{5}\right)} \)
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Para resolver la integral de la función \( F(x) = \left(6 x^{5}+5 x^{4}\right) e^{\left(x^{6}+x^{5}\right)} \) por sustitución, primero identificamos una parte de la función que podemos sustituir. Observamos que el exponente de la función exponencial es \( x^6 + x^5 \).
Definimos la sustitución:
\[
u = x^6 + x^5
\]
Ahora, derivamos \( u \) con respecto a \( x \):
\[
\frac{du}{dx} = 6x^5 + 5x^4
\]
Esto implica que:
\[
du = (6x^5 + 5x^4) \, dx
\]
Ahora podemos reescribir la integral de \( F(x) \):
\[
\int F(x) \, dx = \int \left(6 x^{5}+5 x^{4}\right) e^{\left(x^{6}+x^{5}\right)} \, dx
\]
Sustituyendo \( u \) y \( du \):
\[
\int e^u \, du
\]
La integral de \( e^u \) es simplemente:
\[
e^u + C
\]
Finalmente, sustituimos \( u \) de nuevo en términos de \( x \):
\[
e^{x^6 + x^5} + C
\]
Por lo tanto, la integral de \( F(x) \) es:
\[
\int F(x) \, dx = e^{x^6 + x^5} + C
\]
Quick Answer
La integral de \( F(x) = \left(6 x^{5}+5 x^{4}\right) e^{\left(x^{6}+x^{5}\right)} \) es \( e^{x^6 + x^5} + C \).
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