Question: Let In = $\int _ { 0 } ^ { \pi / 4 } \tan ^ { n }$ x dx. Then I2 + I4<...

Question

Let I_n = $\int _ { 0 } ^ { \pi / 4 } \tan ^ { n }$ x dx. Then I₂ + I₄, I₃ + I₅, I₄ + I₆, I₅ + I₇, …… are in –

Answer 1

We have for r ³ 1,

I_r + I_{r + 2} = $\int _ { 0 } ^ { \pi / 4 }$ tan^r x (1 + tan² x)dx

= $\int _ { 0 } ^ { \pi / 4 }$ tan^r x sec² x dx

= tan^{r + 1} x ] $_ { 0 } ^ { \pi / 4 }$ =

Thus, the given sequence becomes, $\frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 } , \frac { 1 } { 6 } , \ldots$

Which is clearly in H.P.

Question: Let I<sub>n</sub> = \(\int _ { 0 } ^ { \pi / 4 } \tan ^ { n }\) x dx. Then I<sub>2</sub> + I<sub>4<...