AP EAMCET · Maths · Indefinite Integration
\(\int\left(\frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}\right) d x=\)
- A \(\frac{1}{2} \sqrt{1+\mathrm{x}}+\mathrm{c}\)
- B \(\frac{2}{3}(1+\mathrm{x})^{3 / 2}+\mathrm{c}\)
- C \(\sqrt{1+x}+c\)
- D \(2(1+x)^{\frac{3}{2}}+c\)
Answer & Solution
Correct Answer
(B) \(\frac{2}{3}(1+\mathrm{x})^{3 / 2}+\mathrm{c}\)
Step-by-step Solution
Detailed explanation
\( \int\left(\frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}\right) d x=\int\left(\frac{1+x+\sqrt{x}\sqrt{1+x}}{\sqrt{x}+\sqrt{1+x}}\right) d x \) \( =\int\left(\frac{\sqrt{1+x}(\sqrt{1+x}+\sqrt{x})}{\sqrt{x}+\sqrt{1+x}}\right) d x \) \( =\int \sqrt{1+x} \, d x \)…
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