Let \[{I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).\]
If I4+I6 = \[a{\tan ^5}x + b{x^5} + C\], where C is a constant of integration,
then the ordered pair \[\left( {a,b} \right)\] is equal to
Solutions · 1
Official solution
Given,
In = \[\int {{{\tan }^n}x\,dx,\,\,\,n > 1} \]
\[\therefore\,\,\,\] I4 = \[\int {{{\tan }^4}x\,dx} \]
and I6 = \[\int {{{\tan }^6}} x\,dx\]
\[\therefore\,\,\,\] I = I4 + I6
= \[\int {\left( {{{\tan }^4}x + {{\tan }^6}x} \right)} dx\]
= \[\int {{{\tan }^4}} x\left( {1 + {{\tan }^2}x} \right)dx\]
= \[\int {{{\tan }^4}} x.{\sec ^2}x\,dx\]
Let, tanx = t
\[ \Rightarrow \] sec2x dx = dt∴I =∫t4dt=51t5 + C
=51tan5x + C\therefore\,\,\,\[ By comparing with the question, we get
A = \]{1 \over 5}