AP EAMCET · Maths · Indefinite Integration
If \(\int \frac{\mathrm{dx}}{\sin ^3 \mathrm{x}+\cos ^3 \mathrm{x}}=\mathrm{A} \log \left|\frac{\sqrt{2}+\mathrm{t}}{\sqrt{2}-\mathrm{t}}\right|+\mathrm{B} \operatorname{Tan}^{-1}(\mathrm{t})+\mathrm{c}\), then \(\left(\frac{\mathrm{B}}{\mathrm{A}}, \mathrm{t}\right)=\)
- A \((3 \sqrt{2}, \sin x-\cos x)\)
- B \((2 \sqrt{2}, \sin x-\cos x)\)
- C \(\left(\frac{\sqrt{2}}{3}, \sin x-\cos x\right)\)
- D \(\left(\frac{3}{\sqrt{2}}, \sin x+\cos x\right)\)
Answer & Solution
Correct Answer
(B) \((2 \sqrt{2}, \sin x-\cos x)\)
Step-by-step Solution
Detailed explanation
Let \(\mathrm{t} = \sin x - \cos x\). \(\mathrm{dt} = (\cos x + \sin x) \mathrm{dx}\) \(\mathrm{t}^2 = (\sin x - \cos x)^2 = 1 - 2 \sin x \cos x \implies \sin x \cos x = \frac{1-\mathrm{t}^2}{2}\)…
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