COMEDK · Maths · 28. Indefinite Integration
\(\int \dfrac{x d x}{2(1+x)^{3 / 2}}\) is equal to
- A \(\dfrac{x}{\sqrt{1+x}}+C\)
- B \(\dfrac{2+x}{x \sqrt{1+x}}+C\)
- C \(-\dfrac{x}{\sqrt{1+x}}+C\)
- D \(\dfrac{2+x}{\sqrt{1+x}}+C\)
Answer & Solution
Correct Answer
(D) \(\dfrac{2+x}{\sqrt{1+x}}+C\)
Step-by-step Solution
Detailed explanation
Let \(I = \int \dfrac{x}{2(1+x)^{3/2}} dx\). Substitute \(u = 1+x\), so \(du = dx\) and \(x = u-1\). Substituting these into the integral: \(I = \dfrac{1}{2} \int \dfrac{u-1}{u^{3/2}} du = \dfrac{1}{2} \int (u^{-1/2} - u^{-3/2}) du\). Integrating with respect to \(u\):…
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