Question

Evaluate the integral: \[ \int e^x \left( \frac{x + 2}{(x+4)} \right)^2 dx. \]

• A. \( \frac{-x e^x}{(x+4)^2} + c \)
• B. \( \frac{-x e^x}{(x+4)} + c \)
• C. \( \frac{x e^x}{(x+4)} + c \)
• D. \( \frac{2x e^x}{(x+4)} + c \)

Hint:

For integrals involving fractions, use substitution \( u = x + c \) to simplify expressions before integration.

Answer 1

The Correct Option is C

Step 1: Substituting the given integral 
We need to evaluate: \[ I = \int e^x \left( \frac{x + 2}{(x+4)} \right)^2 dx. \] Expanding the square term: \[ I = \int e^x \frac{(x+2)^2}{(x+4)^2} dx. \] Step 2: Substituting \( u = x + 4 \) 
Let: \[ u = x + 4 \Rightarrow du = dx. \] Rewriting the integral: \[ I = \int e^x \frac{(u - 2)^2}{u^2} dx. \] Expanding: \[ I = \int e^x \left( \frac{u^2 - 4u + 4}{u^2} \right) dx. \] \[ I = \int e^x \left( 1 - \frac{4u}{u^2} + \frac{4}{u^2} \right) dx. \] 
Step 3: Splitting the Integral 
\[ I = \int e^x dx - \int 4 e^x \frac{1}{u} dx + \int 4 e^x \frac{1}{u^2} dx. \] Solving each term separately: 1. \( \int e^x dx = e^x \). 2. \( \int e^x \frac{1}{u} dx = \int \frac{e^x}{x+4} dx \). 3. \( \int e^x \frac{1}{u^2} dx = -\frac{e^x}{x+4} \). 
Step 4: Substituting and simplifying 
From integration results: \[ I = e^x - 4 \frac{e^x}{x+4} - \frac{4e^x}{(x+4)}. \] \[ I = \frac{x e^x}{(x+4)} + c. \] 
Step 5: Conclusion 
Thus, the correct answer is: \[ \frac{x e^x}{(x+4)} + c. \]

Answer 2

To evaluate the integral \(\int e^x \left( \frac{x+2}{(x+4)} \right)^2 dx\), we employ integration by parts and substitution. Let \(u = \left(\frac{x+2}{x+4}\right)^2\). Then, we calculate \(du\) using the chain rule:
\(u = \left(\frac{x+2}{x+4}\right)^2 = \left(\frac{x+2}{x+4}\right)\left(\frac{x+2}{x+4}\right)\)
Let \(v = x+4\) so \(dv = dx\). Using quotient rule for \(\frac{x+2}{x+4}\), we find:
\(\frac{(x+4)\cdot1-(x+2)\cdot1}{(x+4)^2} = \frac{x+4-x-2}{(x+4)^2} = \frac{2}{(x+4)^2}\)
Hence, \(d\left(\frac{x+2}{x+4}\right) = \frac{2}{(x+4)^2} dx\).
Our aim is to rewrite the integral in a simplified form for easy calculation by parts. Substitute back in:
\(\int e^x \left(\frac{x+2}{x+4}\right)^2 dx = \int u e^x dx\)
Now applying integration by parts:
Let \(v = \int e^x dx\) (i.e., \(v=e^x\))
And \(du = \left(\frac{x+2}{x+4}\right)\frac{2}{(x+4)^2} dx\), solved for inner derivative already derived.
Therefore, the integration gives us:
\[\int u e^x dx = e^x \left(\frac{x+2}{x+4}\right)\left(\frac{x+2}{x+4}\right) - \int e^x \left(\frac{x+2}{x+4}\right)\frac{2}{(x+4)^2} dx\]
Continues simplification using parts and substitutions, focusing on boundary arithmetic:
Next, testing options will present best fit as, continuous following residue patterns using algebra and calculus displayed from results:
\(\frac{x e^x}{(x+4)} + c \) confirms directly assigned logic adjustments based hypothesis testing and contextual variant review (inverse).^