Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_a^b \frac{9x^2}{1 + 5^x} \, dx$ is equal to:

Step 1: Find the intersection point (a, b)

First, we need to find the coordinates $(a, b)$ where the curve $x^2 = 2y$ and the line $y - 2x - 6 = 0$ intersect. We can substitute the expression for $y$ from the curve equation into the line equation to get a quadratic equation in terms of $x$.

Step 2: Solve the quadratic equation for x

Multiply the equation by 2 to clear the fraction, then rearrange it into standard quadratic form. Factor the quadratic equation to find the possible values for $x$.

Step 3: Determine (a, b) in the second quadrant

Substitute the $x$ values back into $y = \frac{x^2}{2}$ to find the corresponding $y$ values. The problem specifies that the intersection point $(a, b)$ is in the second quadrant, where $x < 0$ and $y > 0$. Therefore, $(a, b) = (-2, 2)$.

Step 4: Set up the integral with limits

Now that we have found $a = -2$ and $b = 2$, we can set up the definite integral with these limits.

Step 5: Use the property of definite integrals

We use the property of definite integrals $\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx$. Here, $a = -2$ and $b = 2$, so $a+b-x = -2+2-x = -x$. Applying this property transforms the integrand.

Step 6: Add the original and transformed integrals

Let the original integral be $I$. We have a second expression for $I$ from the property. Adding these two expressions for $I$ allows us to simplify the integrand significantly, as the denominators become common and the numerators combine to cancel out the $1+5^x$ term.

Step 7: Evaluate the simplified integral

Now, we evaluate the simplified integral $2I = \int_{-2}^{2} 9x^2 \, dx$. We find the antiderivative of $9x^2$, which is $3x^3$, and evaluate it at the limits of integration. Finally, we divide by 2 to find the value of $I$.

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