TS EAMCET · Maths · Definite Integration
\(\int_8^{18} \frac{1}{(x+2) \sqrt{x-3}} d x=\)
- A \(\frac{\pi}{6 \sqrt{5}}\)
- B \(\frac{\pi}{6}\)
- C \(\frac{\pi}{3}\)
- D \(\frac{\pi}{3 \sqrt{5}}\)
Answer & Solution
Correct Answer
(A) \(\frac{\pi}{6 \sqrt{5}}\)
Step-by-step Solution
Detailed explanation
Let \(u = \sqrt{x-3} \Rightarrow x = u^2+3, dx = 2u \, du\). Limits: \(\sqrt{5}\) to \(\sqrt{15}\). \(\int_{\sqrt{5}}^{\sqrt{15}} \frac{2}{(u^2+5)} du\) \(= 2 \left[ \frac{1}{\sqrt{5}} \arctan\left(\frac{u}{\sqrt{5}}\right) \right]_{\sqrt{5}}^{\sqrt{15}}\)…
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