Discussion Forum

Let  $I_{n}=\int_{}^{} tan^{n}x dx (n>1), If $   $I_{4}+I_{6}=a\tan^{5}x+bx^{5}+C,$   , where C is a constant of integration, then  orderd pair (a,b) is equal to

Answer:

Explanation:

We have, $I_{n}=\int_{}^{} tan^{n}x dx$

$\therefore   I_{n}+I_{n+2}=\int_{}^{}\tan^{n}x dx+\int_{}^{}tan^{n}  x dx$

$=\int_{}^{}\tan^{n}x (1+tan^{2}  x) dx$

$=\int_{}^{}\tan^{n}x \sec ^{2}x dx$

$=\frac{\tan^{n+1}x}{n+1}+C$

    Put n=4, we get   $I_{4}+I_{6}=\frac{\tan^{5}x}{5}+C$

  $\therefore  a=\frac{1}{5}$ and $ b=0$