Question

\(\int \frac{10e^x}{(2e^x + 5)^3} dx\) is equal to 
 

• A. \(\frac{5}{2(2e^x + 5)^2} + C\)
• B. \(\frac{-5}{(2e^x + 5)^2} + C\)
• C. \(\frac{-10}{(2e^x + 5)^2} + C\)
• D. \(\frac{-5}{2(2e^x + 5)^2} + C\)
• E. \(\frac{5}{(2e^x + 5)^2} + C\)

Hint:

If the derivative of the base is present in the numerator, the result will always be of the form \(\frac{1}{\text{base}^{n-1}}\) with a negative sign.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The numerator contains the derivative of the expression inside the parentheses in the denominator. This suggests a substitution method.

Step 2: Key Formula or Approach:
Let \(u = 2e^x + 5\). Then \(du = 2e^x dx\).

Step 3: Detailed Explanation:
Let \(u = 2e^x + 5\). Then \(du = 2e^x dx \implies e^x dx = \frac{du}{2}\).
The integral becomes:
\[ \int \frac{10 \cdot \frac{du}{2}}{u^3} = \int 5 u^{-3} du \]
Integrate using the power rule:
\[ = 5 \cdot \frac{u^{-2}}{-2} + C = -\frac{5}{2u^2} + C \]
Substitute back \(u = 2e^x + 5\):
\[ = -\frac{5}{2(2e^x + 5)^2} + C \]

Step 4: Final Answer:
The value is \(\frac{-5}{2(2e^x + 5)^2} + C\).