Question

Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :

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

Answer 1

Answer: (A)

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

Answer 2

$l_{n}=\int \tan ^{n} x d x, n>1$
$l_{4}+l_{6}=\int\left(\tan ^{4} x+\tan ^{6} x\right) d x$
$=\int \tan ^{4} x \sec ^{2} x d x$
Let $\tan x=t$
$\sec ^{2} x d x=d t$
$=\int t^{4} d t$
$=\frac{t^{5}}{5}+C$
$=\frac{1}{5} \tan ^{5} x +C$
$a=\frac{1}{5},\, b=0$