Question

\[ \int \frac{x^3}{(x+1)^2} d x= \]

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

Hint:

Put u = x + 1 so the numerator becomes a polynomial in u, then divide term by term.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The numerator has higher degree than the denominator, so substitute u = x + 1 and divide.

Step 2: Substitute
With \(u=x+1\), \(x^{3}=(u-1)^{3}=u^{3}-3u^{2}+3u-1\). So \(\dfrac{x^{3}}{(x+1)^{2}}=u-3+\dfrac{3}{u}-\dfrac{1}{u^{2}}\).

Step 3: Integrate
\(\int\left(u-3+\dfrac3u-\dfrac1{u^{2}}\right)du=\dfrac{u^{2}}{2}-3u+3\log u+\dfrac1u+c\).

Step 4: Back to x
\(\dfrac{(x+1)^{2}}{2}-3(x+1)=\dfrac{x^{2}}{2}-2x-\dfrac52\). The constant merges into c, giving \(\dfrac{x^{2}}{2}-2x+3\log(x+1)+\dfrac{1}{x+1}+c\).

Final Answer:
The integral is x^2/2 - 2x + 3 log(x + 1) + 1/(x + 1) + c. \[ \boxed{\text{(A) }\dfrac{x^{2}}{2}-2x+3\log(x+1)+\dfrac{1}{x+1}+c} \]