AP EAMCET · Maths · Indefinite Integration
If \(\int \frac{\mathrm{dx}}{1-\sin ^4 \mathrm{x}}=\mathrm{A} \tan \mathrm{x}+\mathrm{B} \operatorname{Tan}^{-1}(\sqrt{2} \tan \mathrm{x})+\mathrm{C}\), then \(\mathrm{A}^2-\mathrm{B}^2=\)
- A \(\frac{1}{2}\)
- B \(\frac{3}{4}\)
- C \(\frac{1}{4}\)
- D \(\frac{1}{8}\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{8}\)
Step-by-step Solution
Detailed explanation
\( \int \frac{\mathrm{dx}}{1-\sin ^4 \mathrm{x}} = \int \frac{\mathrm{dx}}{(1-\sin^2 \mathrm{x})(1+\sin^2 \mathrm{x})} = \int \frac{\mathrm{dx}}{\cos^2 \mathrm{x}(1+\sin^2 \mathrm{x})} \) \( = \int \frac{\sec^2 \mathrm{x}}{1+\sin^2 \mathrm{x}} \mathrm{dx} \) Let…
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