AP EAMCET · Maths · Indefinite Integration
\(\int \frac{x}{\sqrt{x+1}+\sqrt{x-1}} d x=A(x)(x+1)^{\frac{3}{2}}+B(x)(x-1)^{\frac{3}{2}}+C\) , then \(A(x)+B(x)=\)
- A \(\frac{4}{15}\)
- B \(-\frac{4}{15}\)
- C \(\frac{2 x}{5}\)
- D \(-\frac{2 x}{5}\)
Answer & Solution
Correct Answer
(B) \(-\frac{4}{15}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text {Let } I=\int \frac{x}{\sqrt{x+1}+\sqrt{x-1}} d x \\ & =\int \frac{x[\sqrt{x+1}-\sqrt{x-1}]}{x+1-x+1} d x \\ & =\frac{1}{2} \int x \sqrt{x+1} d x-\frac{1}{2} \int x \sqrt{x-1} d x=\frac{1}{2} I_1-\frac{1}{2} I_2 \end{aligned}\) Now,…
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