JEE Mains · Maths · STD 12 - 7.1 indefinite integral
If \(\int \frac{\sin x}{\sin ^{3} x+\cos ^{3} x} d x=\) \(\alpha \log _{\mathrm{e}}|1+\tan \mathrm{x}|+\beta \log _{\mathrm{c}}\left|1-\tan \mathrm{x}+\tan ^{2} \mathrm{x}\right|+\gamma \tan ^{-1}\left(\frac{2 \tan \mathrm{x}-1}{\sqrt{3}}\right)+\mathrm{C}\) when \(\mathrm{C}\) is constant of integration, then the value of \(18\left(\alpha+\beta+\gamma^{2}\right)\) is .... .
- A \(8\)
- B \(1\)
- C \(2\)
- D \(3\)
Answer & Solution
Correct Answer
(D) \(3\)
Step-by-step Solution
Detailed explanation
\(=\int \frac{\frac{\sin x}{\cos ^{3} x}}{1+\tan ^{3} x} d x=\int \frac{\tan x \cdot \sec ^{2} x}{(\tan x+1)\left(1+\tan ^{2} x-\tan x\right)} \,d x\) Let \(\tan x=t \Rightarrow \sec ^{2} x \cdot \,d x=d t\) \(=\int \frac{t}{(t+1)\left(t^{2}-t+1\right)}\, d t\)…
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