int frac{(x+3) e^{x}}{(x+4)^{2}} d x is equ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We are given the integral $ I = \int \frac{(x+3) e^{x}}{(x+4)^{2}} d x $. To solve this,we rewrite the numerator as $(x+4-1)$: $ I = \int e^{x} \f

Vedclass · Accepted Answer

$ \frac{e^{x}}{(x+4)}+C $

Vedclass · Answer

(C) We are given the integral $ I = \int \frac{(x+3) e^{x}}{(x+4)^{2}} d x $. To solve this,we rewrite the numerator as $(x+4-1)$: $ I = \int e^{x} \frac{(x+4-1)}{(x+4)^{2}} d x $. Splitting the fraction,we get: $ I = \int e^{x} \left[ \frac{x+4}{(x+4)^{2}} - \frac{1}{(x+4)^{2}} \right] d x $. $ I = \int e^{x} \left[ \frac{1}{x+4} - \frac{1}{(x+4)^{2}} \right] d x $. We know the standard integration formula: $ \int e^{x} (f(x) + f'(x)) d x = e^{x} f(x) + C $. Here,let $ f(x) = \frac{1}{x+4} $. Then,$ f'(x) = -\frac{1}{(x+4)^{2}} $. Substituting these into the formula,we get: $ I = e^{x} \left( \frac{1}{x+4} \right) + C = \frac{e^{x}}{x+4} + C $.

$ \int \frac{(x+3) e^{x}}{(x+4)^{2}} d x $ is equal to

Similar Questions

$ \int e^{x}\left(\frac{1+\sin x}{1+\cos x}\right) d x $ is

$\int \left( \frac{1-\log x}{1+(\log x)^2} \right)^2 dx = $

If an antiderivative of $f(x)$ is $e^x$ and that of $g(x)$ is $\cos x,$ then $\int f(x) \cos x \, dx + \int g(x) e^x \, dx = $

$\int e^{4x}(\sin 3x - \cos 3x) dx = $

$\int e^x \left( \frac{\sec^2 x + \tan x - \cot x}{\sin x} \right) dx =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module