int e^x left( frac{x+5}{(x+6)^2} right) dx is | vedclass

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Question

(D) We use the standard integral formula: $\int e^x [f(x) + f'(x)] dx = e^x f(x) + c$. First,rewrite the integrand: $\frac{x+5}{(x+6)^2} = \frac{(x+6)

Vedclass · Accepted Answer

$\frac{e^x}{x+6} + c$,where $c$ is the constant of integration.

Vedclass · Answer

(D) We use the standard integral formula: $\int e^x [f(x) + f'(x)] dx = e^x f(x) + c$. First,rewrite the integrand: $\frac{x+5}{(x+6)^2} = \frac{(x+6) - 1}{(x+6)^2} = \frac{1}{x+6} - \frac{1}{(x+6)^2}$. Let $f(x) = \frac{1}{x+6}$. Then $f'(x) = -\frac{1}{(x+6)^2}$. Substituting these into the integral: $\int e^x \left( \frac{1}{x+6} - \frac{1}{(x+6)^2} \right) dx = \int e^x [f(x) + f'(x)] dx$. Applying the formula,we get $e^x f(x) + c = \frac{e^x}{x+6} + c$.

$\int e^x \left( \frac{x+5}{(x+6)^2} \right) dx$ is equal to:

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