Q52 / 160 · AP EAMCET 2020 (21 Sep Shift 2) · Mathematics

\(\int x\left(\tan ^2 x\right) d x=\)

  • Correct answer
    A
    \(x \tan (x)-\log _e(\sec x)-\frac{x^2}{2}+c\)
  • B
    \(x \tan (x)+\log _e(\sec x)-\frac{x^2}{2}+c\)
  • C
    \(x \tan (x)-\log _e(\sec x)+\frac{x^2}{2}+c\)
  • D
    \(x \tan (x)+\log _e(\sec x)+\frac{x^2}{2}+c\)

Solution

\(\begin{aligned} I & =\int x\left(\tan ^2 x\right) d x=x \int\left(\sec ^2 x-1\right) d x -\int\left(1 \int\left(\sec ^2 x-1\right) d x\right) d x \\ & =x(\tan x-x)-\int(\tan x-x) d x \\ & =x \tan x-x^2-\log _e(\sec x)+\frac{x^2}{2}+C \\ & =x \tan x-\log _e(\sec x)-\frac{x^2}{2}+C \end{aligned}\) Hence, option (a) is correct.

Hence, the correct answer is option: 'A'.

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