JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(\int_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} d x=\alpha \sqrt{2}+\beta \sqrt{3}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to.
- A \(10\)
- B \(11\)
- C \(12\)
- D \(13\)
Answer & Solution
Correct Answer
(A) \(10\)
Step-by-step Solution
Detailed explanation
Put \(1+ x ^{2}= t ^{2}\) \(2 x dx =2 t dt\) \(X dx = t d t\) \(\therefore \int_{1}^{2} \frac{15\left( t ^{2}-1\right) t dt }{\sqrt{ t ^{2}+ t ^{3}}}\) \(15 \int_{1}^{2} \frac{ t \left( t ^{2}-1\right)}{ t \sqrt{1+ t }} dt\) Put \(1+ t = u ^{2}\) \(dt =2 u du\)…
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