MHT CET · Maths · Indefinite Integration
\(\int \frac{d x}{1+\sqrt{x}}=\)
- A \(2 \sqrt{x}-2 \log |1+\sqrt{x}|+c\)
- B \(\sqrt{x}+\log |1+\sqrt{x}|+c\)
- C \(2 \sqrt{x}+\log |1+\sqrt{x}|+c\)
- D \(\sqrt{x}-\log |1+\sqrt{x}|+c\)
Answer & Solution
Correct Answer
(A) \(2 \sqrt{x}-2 \log |1+\sqrt{x}|+c\)
Step-by-step Solution
Detailed explanation
(C)
Let \(\quad \mathrm{I}=\int \frac{\mathrm{dx}}{1+\sqrt{\mathrm{x}}}\)
Put \(\quad \sqrt{\mathrm{x}}=\mathrm{t} \Rightarrow \frac{1}{2 \sqrt{\mathrm{x}}} \mathrm{dx}=\mathrm{dt} \Rightarrow \mathrm{dx}=2 \mathrm{t} \mathrm{dt}\)
\(\therefore \mathrm{I}=\int \frac{2 \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}\)
\(\quad=2 \int \frac{(\mathrm{t}+1)-1}{1+\mathrm{t}} \mathrm{dt}=2 \int \mathrm{dt}-2 \int \frac{\mathrm{dt}}{1+\mathrm{t}}\)
\(\quad=2 \mathrm{t}-2 \log |1+\mathrm{t}|=2 \sqrt{\mathrm{x}}-2 \log |1+\sqrt{\mathrm{x}}|+\mathrm{c}\)
Let \(\quad \mathrm{I}=\int \frac{\mathrm{dx}}{1+\sqrt{\mathrm{x}}}\)
Put \(\quad \sqrt{\mathrm{x}}=\mathrm{t} \Rightarrow \frac{1}{2 \sqrt{\mathrm{x}}} \mathrm{dx}=\mathrm{dt} \Rightarrow \mathrm{dx}=2 \mathrm{t} \mathrm{dt}\)
\(\therefore \mathrm{I}=\int \frac{2 \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}\)
\(\quad=2 \int \frac{(\mathrm{t}+1)-1}{1+\mathrm{t}} \mathrm{dt}=2 \int \mathrm{dt}-2 \int \frac{\mathrm{dt}}{1+\mathrm{t}}\)
\(\quad=2 \mathrm{t}-2 \log |1+\mathrm{t}|=2 \sqrt{\mathrm{x}}-2 \log |1+\sqrt{\mathrm{x}}|+\mathrm{c}\)
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