The minimum odd value of 'a' `(agt1)` for which `int_(10)^(19)(sinx)/(1+x^(a))dxlt(1)/(9),` is equal to

Options

A. 1
B. 3
C. 5
D. 9

Answer

To solve the problem, we need to find the minimum odd value of \( a \) (where \( a > 1 \)) such that

\[ \int_{10}^{19} \frac{\sin x}{1 + x^a} \, dx < \frac{1}{9}. \]

### Step-by-Step Solution:

1. **Set Up the Integral**:
We start with the integral:
\[ I(a) = \int_{10}^{19} \frac{\sin x}{1 + x^a} \, dx. \]

2. **Use Inequalities**:
We know that \( |\sin x| \leq 1 \) for all \( x \). Therefore, we can write:
\[ \left| \frac{\sin x}{1 + x^a} \right| \leq \frac{1}{1 + x^a}. \]
This leads to:
\[ I(a) < \int_{10}^{19} \frac{1}{1 + x^a} \, dx. \]

3. **Estimate the Integral**:
We can further estimate:
\[ \frac{1}{1 + x^a} < \frac{1}{x^a} \quad \text{for } x \geq 10. \]
Thus, we have:
\[ I(a) < \int_{10}^{19} \frac{1}{x^a} \, dx. \]

4. **Calculate the Integral**:
Now we compute the integral:
\[ \int \frac{1}{x^a} \, dx = \frac{x^{-a + 1}}{-a + 1} = \frac{-1}{a - 1} x^{-a + 1}. \]
Evaluating from 10 to 19 gives:
\[ \int_{10}^{19} \frac{1}{x^a} \, dx = \left[ \frac{-1}{a - 1} \left(19^{-a + 1} - 10^{-a + 1}\right) \right]. \]

5. **Set Up the Inequality**:
We want:
\[ \frac{-1}{a - 1} \left(19^{-a + 1} - 10^{-a + 1}\right) < \frac{1}{9}. \]

6. **Simplify the Inequality**:
This can be rearranged to:
\[ 19^{-a + 1} - 10^{-a + 1} > -\frac{a - 1}{9}. \]

7. **Trial Values for \( a \)**:
We need to find the minimum odd value of \( a > 1 \). We will try \( a = 3 \):
\[ I(3) = \int_{10}^{19} \frac{1}{x^3} \, dx. \]
Evaluating this integral:
\[ = \left[ \frac{-1}{2} \left(19^{-2} - 10^{-2}\right) \right] = \frac{-1}{2} \left(\frac{1}{361} - \frac{1}{100}\right). \]
Calculating this gives:
\[ = \frac{-1}{2} \left(\frac{100 - 361}{36100}\right) = \frac{261}{72200} \approx 0.0036. \]

8. **Check the Condition**:
We check if \( \frac{261}{72200} < \frac{1}{9} \):
\[ \frac{1}{9} \approx 0.1111. \]
Since \( 0.0036 < 0.1111 \), the condition holds.

9. **Conclusion**:
Therefore, the minimum odd value of \( a \) for which the integral is less than \( \frac{1}{9} \) is:
\[ \boxed{3}. \]