Question

Evaluate the following integrals:

(a) ∫ x² ln(x² + 9) dx
(b) ∫[x e^(4x)] dx from -1/4 to 0
(c) ∫[sin³(2x) cos⁵(2x)] dx

Answer 1

(a) 1/18 * (x² + 9)³ * ln(x² + 9) - 1/2 * x² - C, where C is the constant of integration.

(b) 1/(4e)

(c) 1/32 * cos⁶(2x) - 1/18 * cos⁴(2x) + 1/36 * cos²(2x) + C, where C is the constant of integration.

Step-by-step explanation:

**(a)**

To evaluate ∫ x² ln(x² + 9) dx, we use integration by parts. Let u = ln(x² + 9) and dv = x² dx. Applying the integration by parts formula ∫ u dv = uv - ∫ v du, we find the antiderivative and simplify to get the final answer.

**(b)**

For ∫[x e^(4x)] dx from -1/4 to 0, we use the fundamental theorem of calculus and evaluate the antiderivative at the upper and lower limits of integration. Subtracting the lower limit from the upper limit gives the final result.

**(c)**

To compute ∫[sin³(2x) cos⁵(2x)] dx, we apply the power reduction formula for sine and cosine, simplify the expression, and then integrate. The final result includes the constant of integration, denoted by C, which accounts for the family of antiderivatives.