Question

Evaluate the integral ∫(7x²)/(a² - x²) dx from 0

Answer 1

To evaluate the integral of 7x²/(a² - x²) without specific limits, use partial fractions decomposition, expressing the integrand as the sum of two simpler rational functions and then integrating each part separately.

Step-by-step explanation:

To evaluate the integral ∫(7x²)/(a² - x²) dx, we typically would look for a method such as substitution or partial fractions, depending on whether 'a' is just a constant or a function of 'x'. However, since we do not have the limits of integration, we can only express the indefinite integral in a general form. For this particular integral, if 'a' is a non-zero constant, the method of partial fractions would be appropriate.

We start by factorizing the denominator:

• a² - x² = (a + x)(a - x)

Then, we would express our integrand as:

• 7x²/(a + x)(a - x)

Now we would look for two constants A and B such that:

• 7x² = A(a - x) + B(a + x)

After finding A and B, we split the integral into two parts and integrate each one separately:

1. ∫ A/(a - x) dx
2. ∫ B/(a + x) dx

This process is known as partial fractions decomposition, and it simplifies the integration of rational functions.