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Question

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


A
(15,1)
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B
(15,0)
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C
(15,1)
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D
(15,0)
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Solution

The correct option is C $$\left(\displaystyle\frac{1}{5}, 0\right)$$
Given that: $$\displaystyle \int \tan ^n x  dx$$

Using induction formula for $$\tan$$ we get:

$$I_n = \dfrac{\tan^ {(n -1)}x}{(n-1)} - I _{(n-2)} +c$$

$$ I_n + I_{(n-2)} = \dfrac{tan^{n-1}x }{(n-1)}+c$$
Hence $$ I_6+ I_4 = \dfrac{\tan^{6-1}x}{6-1} +c$$
             $$ = \dfrac{\tan ^5 x}{5} + 0x+c$$
$$\therefore a = \dfrac{1}{5}, b=0$$



Mathematics

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