Questions: Online Homework 10: Problem 2 (1 point) Consider the integral [ int0^1 frac9sqrt1-x^2 d x ] If the integral is divergent, type an upper-case "D". Otherwise, evaluate the integral.

Solution Steps

To solve the given integral, we recognize that the integrand \(\frac{9}{\sqrt{1-x^{2}}}\) resembles the derivative of the arcsine function. Specifically, the integral of \(\frac{1}{\sqrt{1-x^{2}}}\) is \(\arcsin(x)\). Therefore, we can use this property to evaluate the integral.

1. Identify the integral as a form of the arcsine function.
2. Use the antiderivative of \(\frac{1}{\sqrt{1-x^{2}}}\), which is \(\arcsin(x)\).
3. Multiply by the constant factor (9 in this case).
4. Evaluate the definite integral from 0 to 1.

We are tasked with evaluating the integral

\[ \int_{0}^{1} \frac{9}{\sqrt{1-x^{2}}} \, dx. \]

The integrand \(\frac{1}{\sqrt{1-x^{2}}}\) is known to have the antiderivative

\[ \arcsin(x). \]

Thus, we can express the integral as

\[ 9 \int_{0}^{1} \frac{1}{\sqrt{1-x^{2}}} \, dx = 9 \left[ \arcsin(x) \right]_{0}^{1}. \]

Now we evaluate the antiderivative at the bounds:

\[ \arcsin(1) - \arcsin(0) = \frac{\pi}{2} - 0 = \frac{\pi}{2}. \]

Multiplying by 9 gives us:

\[ 9 \cdot \frac{\pi}{2} = \frac{9\pi}{2}. \]

Final Answer