JEE Mains · Maths · STD 12 - 7.1 indefinite integral
Let \(f(x)=\int \frac{d x}{\left(3+4 x^2\right) \sqrt{4-3 x^2}},|x| < \frac{2}{\sqrt{3}}\). If \(f(0)=0\) and \(f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right), \alpha, \beta > 0\), then \(\alpha^2+\beta^2\) is equal to \(.........\).
- A \(28\)
- B \(26\)
- C \(25\)
- D \(24\)
Answer & Solution
Correct Answer
(A) \(28\)
Step-by-step Solution
Detailed explanation
\(f(x)=\int \frac{d x}{\left(3+4 x^2\right) \sqrt{4-3 x^2}}\) \(x=\frac{1}{t}\) \(=\int \frac{\frac{-1}{t^2} d t}{\frac{\left(3 t^2+4\right)}{t^2} \frac{\sqrt{4 t^2-3}}{t}}\) \(=\int \frac{-d t \cdot t}{\left(3 t^2+4\right) \sqrt{4 t^2-3}}: \text { Put } 4 t^2-3=z^2\)…
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