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

Options

A. `(-(1)/(5),0)`
B. `(-(1)/(5),1)`
C. `((1)/(5),0)`
D. `((1)/(5),-1)`

Explanation

To solve the problem, we need to find the ordered pair \((a, b)\) such that:

\[ I_4 + I_6 = a \tan^5 x + b x^5 + C \]

where \(I_n = \int \tan^n x \, dx\) for \(n > 1\).

### Step-by-step Solution:

1. **Write the expression for \(I_4 + I_6\)**:
\[ I_4 + I_6 = \int \tan^4 x \, dx + \int \tan^6 x \, dx \]

2. **Combine the integrals**:
\[ I_4 + I_6 = \int (\tan^4 x + \tan^6 x) \, dx \]

3. **Factor out \(\tan^4 x\)**:
\[ I_4 + I_6 = \int \tan^4 x (1 + \tan^2 x) \, dx \]

4. **Use the identity \(1 + \tan^2 x = \sec^2 x\)**:
\[ I_4 + I_6 = \int \tan^4 x \sec^2 x \, dx \]

5. **Substitute \(t = \tan x\)**, then \(dt = \sec^2 x \, dx\):
\[ I_4 + I_6 = \int t^4 \, dt \]

6. **Integrate \(t^4\)**:
\[ \int t^4 \, dt = \frac{t^5}{5} + C = \frac{\tan^5 x}{5} + C \]

7. **Rewrite the expression**:
\[ I_4 + I_6 = \frac{\tan^5 x}{5} + C \]

8. **Compare with the given format**:
\[ I_4 + I_6 = a \tan^5 x + b x^5 + C \]
From this, we can see that:
- \(a = \frac{1}{5}\)
- \(b = 0\)

9. **Final ordered pair**:
\[ (a, b) = \left(\frac{1}{5}, 0\right) \]

### Final Answer:
The ordered pair \((a, b)\) is \(\left(\frac{1}{5}, 0\right)\).