COMEDK · Maths · 28. Indefinite Integration

\(\int \frac{x d x}{2(1+x)^{3 / 2}}\) is equal to

A \(\frac{2+x}{\sqrt{1+x}}+C\)
B \(\frac{2+x}{x \sqrt{1+x}}+C\)
C \(\frac{x}{\sqrt{1+x}}+C\)
D \(-\frac{x}{\sqrt{1+x}}+C\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{2+x}{\sqrt{1+x}}+C\)

Step-by-step Solution

Detailed explanation

Let \(I=\int \frac{x d x}{2(1+x)^{3 / 2}}\) On putting, \(1+x=t\), we get \(d x=d t\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From COMEDK

Match the ions given in Column I with their characteristic properties given in Column II.

S.No.	Column I	S.No.	Column II
A	\(Cr^{2+}\)	P	Stable in aqueous medium and has magnetic moment equal to 5.92 BM.
B	\(Cu^+\)	Q	Only stable oxidation state and does not act as reducing agent.
C	\(Mn^{2+}\)	R	Is a good reducing agent due to attainment of stable \(d^3\) configuration.
D	\(Sc^{3+}\)	S	Unstable in aqueous medium and readily disproportionates.

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