int frac{e^x(x + 3)}{(x + 5)^3} dx = | vedclass

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(A) We are given the integral $I = \int \frac{e^x(x + 3)}{(x + 5)^3} dx$. First,we rewrite the numerator $(x + 3)$ as $(x + 5 - 2)$: $I = \int \frac{e

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$\frac{e^x}{(x + 5)^2} + c$

Vedclass · Answer

(A) We are given the integral $I = \int \frac{e^x(x + 3)}{(x + 5)^3} dx$. First,we rewrite the numerator $(x + 3)$ as $(x + 5 - 2)$: $I = \int \frac{e^x(x + 5 - 2)}{(x + 5)^3} dx$ $I = \int e^x \left[ \frac{x + 5}{(x + 5)^3} - \frac{2}{(x + 5)^3} \right] dx$ $I = \int e^x \left[ \frac{1}{(x + 5)^2} - \frac{2}{(x + 5)^3} \right] dx$ We use the standard integration formula $\int e^x (f(x) + f'(x)) dx = e^x f(x) + c$. Let $f(x) = \frac{1}{(x + 5)^2} = (x + 5)^{-2}$. Then $f'(x) = -2(x + 5)^{-3} = -\frac{2}{(x + 5)^3}$. Since the integrand is in the form $e^x(f(x) + f'(x))$,the integral is $e^x f(x) + c$. Thus,$I = \frac{e^x}{(x + 5)^2} + c$.

$\int \frac{e^x(x + 3)}{(x + 5)^3} dx = $

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