Question

Let $I_n=\int {\tan }^{n}xdx,(n>1)$. If $I_4+I_6=a{\tan }^{5}x+b{x}^5+C$, where $C$ is a constant of integration, then the ordered pair $(a,b)$ is equal to

• A. $(-\frac{1}{5},1)$
• B. $(\frac{1}{5},0)$
• C. $(\frac{1}{5},-1)$
• D. $(-\frac{1}{5},0)$

Answer 1

Answer: (B)

$(\frac{1}{5},0)$

$I_4+I_6=\int (ta{n}^{4}x+ta{n}^{6}x)dx$

$=\int ta{n}^{4}x\left(se{c}^{2}x\right)dx$

$tanx=t$

$se{c}^{2}xdx=dt$

$=\int t^{4}dt$

$=\frac{t^5}{5}+c$

$=\frac{ta{n}^{5x}}{5}+c$

$a=1/5,b=0$