AP EAMCET · Maths · Indefinite Integration
If \(\int \frac{d x}{\cos ^4 x+\sin ^4 x}=\frac{1}{\sqrt{2}} \tan ^{-1}[g(x)]+C\), then \(g(x)\) equals
- A \(\frac{\tan x-\cot x}{\sqrt{2}}\)
- B \(\frac{\tan x+\cot x}{\sqrt{2}}\)
- C \(\frac{\sin x-\cos x}{\sqrt{2}}\)
- D \(\frac{\sin x+\cos x}{\sqrt{2}}\)
Answer & Solution
Correct Answer
(A) \(\frac{\tan x-\cot x}{\sqrt{2}}\)
Step-by-step Solution
Detailed explanation
\[ \begin{aligned} & \text { (a) } I=\int \frac{d x}{\cos ^4 x+\sin ^4 x} \\ & =\int \frac{\sec ^4 x}{1+\tan ^4 x} d x=\int \frac{\sec ^2 x\left(1+\tan ^2 x\right)}{1+\tan ^4 x} d x \end{aligned} \] Let \(\tan x=t\), then \(\sec ^2 x d x=d t\)…
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