Question

\[ \int \mathrm{e}^x \frac{(x-1)}{(x+1)^3} \mathrm{~d} x= \]

• A. \(\mathrm{e}^{x}(x+1)^{2}+\mathrm{c}\), where c is the constant of integration
• B. \(\mathrm{e}^{x}(x+1)^{3}+\mathrm{c}\), where c is the constant of integration
• C. \(\frac{\mathrm{e}^{x}}{(x+1)^{2}}+\mathrm{c}\), where c is the constant of integration
• D. \(\frac{\mathrm{e}^{x}}{(x+1)^{3}}+\mathrm{c}\), where c is the constant of integration

Hint:

Split the fraction as 1/(x+1)^2 - 2/(x+1)^3 and use the form e^x (f + f') dx.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Write \(x-1=(x+1)-2\) to split the fraction.

Step 2: Split
\(\frac{x-1}{(x+1)^3}=\frac{1}{(x+1)^2}-\frac{2}{(x+1)^3}\).

Step 3: Use the standard form
Let \(f(x)=\frac{1}{(x+1)^2}\). Then \(f'(x)=-\frac{2}{(x+1)^3}\), so the integrand is \(e^x\left(f+f'\right)\).

Step 4: Integrate
\(\int e^x(f+f')dx=e^xf(x)+c=\frac{e^x}{(x+1)^2}+c\).

Final Answer:
The integral is \(\frac{e^x}{(x+1)^2}+c\). \[ \boxed{\text{(C) }\dfrac{\mathrm{e}^{x}}{(x+1)^{2}}+\mathrm{c}\text{, where c is the constant of integration}} \]