Q110 / 160 · AP EAMCET 2023 (18 May Shift 1) · Mathematics

If $I_n=\int \tan ^n x d x(n>1)$, then $I_4+I_6=$

  • Correct answer
    A
    $\frac{1}{5} \tan ^5 x+C$
  • B
    $-\frac{1}{5} \tan ^5 x+C$
  • C
    $\frac{1}{10} \tan ^5 x+C$
  • D
    $-\frac{1}{10} \tan ^5 x+C$

Solution

$\int \tan ^n x d x$ $\begin{aligned} & I_4+I_6=\int \tan ^4 x d x+\int \tan ^6 x d x \\ & =\int\left(\tan ^4 x+\tan ^6 x\right) d x=\int \tan ^4 x \sec ^2 x d x\end{aligned}$ Let $\tan x=t \Rightarrow \sec ^2 x d x=d t$ $\therefore \quad \mathrm{I}_4+\mathrm{I}_6=\int \mathrm{t}^4 \mathrm{dt}=\frac{\mathrm{t}^5}{5}+\mathrm{C}$ $\mathrm{I}_4+\mathrm{I}_6=\frac{1}{5} \tan ^5 \mathrm{x}+\mathrm{C}$

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

Relevant Questions