MHT CET · Maths · Indefinite Integration
\(\int \frac{e^{x}}{\sqrt{x}}(1+2 x) d x=\)
- A \(\frac{1}{\sqrt{x}} e^{x}+c\)
- B \(2 \sqrt{x} e^{x}+c\)
- C \(\frac{\sqrt{x}}{2} e^{x}+c\)
- D \(\sqrt{x} e^{x}+c\)
Answer & Solution
Correct Answer
(B) \(2 \sqrt{x} e^{x}+c\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{e^{x}}{\sqrt{x}}(1+2 x) d x\)
\(=\int e^{x}\left(\frac{1}{\sqrt{x}}+2 \sqrt{x}\right) d x=\int e^{x}\left(2 \sqrt{x}+\frac{1}{\sqrt{x}}\right) d x\) \(=2 \int e^{x}\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x\)
\(=2 e^{x} \sqrt{x}+c\)
\(=\int e^{x}\left(\frac{1}{\sqrt{x}}+2 \sqrt{x}\right) d x=\int e^{x}\left(2 \sqrt{x}+\frac{1}{\sqrt{x}}\right) d x\) \(=2 \int e^{x}\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x\)
\(=2 e^{x} \sqrt{x}+c\)
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