Question

If $I_n={\int }_0^{\pi /4}{\tan }^{n}x\times {\sec }^{2}xdx,$ then $I_1,I_2,I_3,$ .......are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. none

Answer 1

Answer: (C)

H.P.

$I_n={\int }_0^{\pi /4}{\tan }^{n}x{\sec }^{2}xdx$
Let, $t=\tan x$ $\Rightarrow {\sec }^{2}xdx=dt$
Therefore, $I_n={\int }_0^1{t}^{n}dt={\left(\frac{t}{n+1}\right)}_0^1=\frac{1}{n}$
Therefore, $I_1,I_2,I_3\cdots$ are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots$ which are in H.P
Ans: C