int {{e^x}left[ {f(x) + f'(x)} right],dx} is | vedclass

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Question

(A) We are given the integral $I = \int e^x [f(x) + f'(x)] \, dx$. Using the distributive property of integration,we can write this as: $I = \int e^x

Vedclass · Accepted Answer

${e^x}f(x) + C$

Vedclass · Answer

(A) We are given the integral $I = \int e^x [f(x) + f'(x)] \, dx$. Using the distributive property of integration,we can write this as: $I = \int e^x f(x) \, dx + \int e^x f'(x) \, dx$. Now,apply integration by parts to the first integral $\int e^x f(x) \, dx$,taking $f(x)$ as the first function and $e^x$ as the second function: $\int e^x f(x) \, dx = f(x) \int e^x \, dx - \int \left( f'(x) \int e^x \, dx \right) \, dx$. $= f(x) e^x - \int f'(x) e^x \, dx$. Substituting this back into the expression for $I$: $I = [f(x) e^x - \int e^x f'(x) \, dx] + \int e^x f'(x) \, dx$. $I = e^x f(x) + C$. Thus,the correct option is $A$.

$\int {{e^x}\left[ {f(x) + f'(x)} \right]\,dx} $ is equal to

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