AP EAMCET · Maths · Indefinite Integration
\[
\int\left(\sec ^4 x+\tan ^4 x\right) d x=
\]
- A \(\frac{2}{3} \tan ^3 x-\frac{2}{3} \tan x+x+c\)
- B \(\frac{1}{3} \sec ^2 x \tan x+\frac{5}{3} \tan x+\frac{\tan ^3 x}{3}+x+c\)
- C \(\frac{2}{3} \tan ^3 x+x+c\)
- D \(\frac{1}{3} \sec ^2 x \tan x-\frac{5}{3} \tan x+\frac{\tan ^3 x}{3}+x+c\)
Answer & Solution
Correct Answer
(C) \(\frac{2}{3} \tan ^3 x+x+c\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { } I=\int\left(\sec ^4 x+\tan ^4 x\right) d x \\ & =\int\left[\sec ^4 x+\left(\sec ^2 x-1\right)^2\right] d x \\ & =\int\left(2 \sec ^4 x-2 \sec ^2 x+1\right) d x \\ & =\int\left[2\left(1+\tan ^2 x\right) \sec ^2 x-2 \sec ^2 x+1\right] d x \\ &…
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