If `I_(1) =int _(0)^(1) (1+x ^(8))/(1+x^(4)) dx sand I_(2)=int _(0)^(1) (1+ x ^(9))/(1+x ^(2)) dx,` then:

Options

A. `I_(1)gt 1 ,I_(2)lt 1`
B. `I _(1) lt 1, I_(2) gt 1`
C. `1 lt I_(1) lt I_(2)`
D. `I_(2) lt I_(1) lt 1`

Answer

To solve the problem, we need to compare the two integrals \( I_1 \) and \( I_2 \):

\[ I_1 = \int_0^1 \frac{1 + x^8}{1 + x^4} \, dx \]

\[ I_2 = \int_0^1 \frac{1 + x^9}{1 + x^2} \, dx \]

### Step 1: Analyze the integrands

We can analyze the integrands of both integrals. Since \( x \) lies between 0 and 1, we can observe the behavior of the terms \( x^8, x^9, x^4, \) and \( x^2 \).

- For \( 0 \leq x < 1 \):
- \( x^9 < x^8 \)
- \( x^4 < x^2 \)