Home › MCQs › EEJ MAIN Mathematics › Question #6972
Back to Questions
EEJ MAIN Mathematics QUESTION #6972
Question 1

Let $I_n=\displaystyle\int\tan^n x\,dx$ for $n>1$. If $I_4+I_6 = a\tan^5x + bx^5 + C$, find the ordered pair $(a,b)$.

  • $\left(-\dfrac{1}{5},1\right)$
  • $\left(\dfrac{1}{5},0\right)$✔️
  • $\left(\dfrac{1}{5},-1\right)$
  • $\left(-\dfrac{1}{5},0\right)$
Correct Answer Explanation

Use the reduction: $I_n+I_{n-2}=\displaystyle\int\tan^{n-2}x\cdot\tan^2x\,dx+\int\tan^{n-2}x\,dx=\int\tan^{n-2}x\sec^2x\,dx=\dfrac{\tan^{n-1}x}{n-1}+C$

$I_4+I_6$: set $n=5$: $I_4+I_6=\dfrac{\tan^5x}{5}+C$

Comparing with $a\tan^5x+bx^5+C$: $a=\dfrac{1}{5}$, $b=0$.

$(a,b) = \left(\dfrac{1}{5},0\right)$