Question

If $I_n = \int \tan^n x \, dx \quad (n>1)$, then $I_4 + I_6 =$

• A. $\dfrac{1}{5} \tan^5 x + C$
• B. $-\dfrac{1}{5} \tan^5 x + C$
• C. $\dfrac{1}{10} \tan^5 x + C$
• D. $-\dfrac{1}{10} \tan^5 x + C$

Hint:

Reduction Formula for $\int \tan^n x \, dx$: \[ I_n + I_n-2 = \frac\tan^n-1 xn - 1 \] Apply recursively or directly plug values to compute combinations.

Answer 1

The Correct Option is A

Use reduction formula: \[ I_n = \frac{\tan^{n-1} x}{n - 1} - I_{n-2} \Rightarrow I_n + I_{n-2} = \frac{\tan^{n-1} x}{n - 1} \] Apply for $n = 6$: \[ I_6 + I_4 = \frac{\tan^5 x}{5} \Rightarrow I_4 + I_6 = \boxed{\frac{1}{5} \tan^5 x + C} \]