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Question: If \(I _ { 8 } + I _ { 6 }\) equals....

If I8+I6I _ { 8 } + I _ { 6 } equals.

A

14\frac { 1 } { 4 }

B

15\frac { 1 } { 5 }

C

16\frac { 1 } { 6 }

D

17\frac { 1 } { 7 }

Answer

17\frac { 1 } { 7 }

Explanation

Solution

In=∫0π/4(sec⁡2θ−1)tan⁡n−2θdθI _ { n } = \int _ { 0 } ^ { \pi / 4 } \left( \sec ^ { 2 } \theta - 1 \right) \tan ^ { n - 2 } \theta d \theta

In=∫0π/4sec⁡2θtan⁡n−2θdθ−∫0π/2tan⁡n−2θdθI _ { n } = \int _ { 0 } ^ { \pi / 4 } \sec ^ { 2 } \theta \tan ^ { n - 2 } \theta d \theta - \int _ { 0 } ^ { \pi / 2 } \tan ^ { n - 2 } \theta d \theta

In=[tan⁡n−1θn−1]0π/4−In−2⇒In+In−2=1n−1I _ { n } = \left[ \frac { \tan ^ { n - 1 } \theta } { n - 1 } \right] _ { 0 } ^ { \pi / 4 } - I _ { n - 2 } \Rightarrow I _ { n } + I _ { n - 2 } = \frac { 1 } { n - 1 }

Hence I8+I6=18−1=17I _ { 8 } + I _ { 6 } = \frac { 1 } { 8 - 1 } = \frac { 1 } { 7 }