Question

Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration Do not evatuate the integral

∫ 5x² + 18x + 20/(2x + 3)(x² + 4x + 8) dx

A. Partial fractions
B. Substitution with x= (2x + 3)
C. Use long division, and then apply partial fractions
D. Integration by parts

Answer 1

To evaluate the given integral, we can use the technique of partial fractions. First, we need to simplify the integrand by factoring the denominator. Then, apply partial fractions to decompose the integrand and integrate each term separately.

Step-by-step explanation:

The given integral is: ∫(5x² + 18x + 20) / ((2x + 3)(x² + 4x + 8)) dx

To evaluate this integral, we can use the technique of partial fractions. First, we need to simplify the integrand by factoring the denominator.

Factoring the denominator, we get: (2x + 4)(x² + 4x + 8)

Now, we can apply partial fractions to decompose the integrand. Let's express the integrand as a sum of two fractions, with the denominators being the individual factors of the factored denominator.

After simplifying the integrand using partial fractions, we can integrate each term separately. Once integrated, we combine the results to get the final solution.

Therefore answer is A. Partial fractions.