Question

Evaluate the integral using integration by parts where possible. HINT (See Examples 1-3.] (Use C for the constant of integration.)

∫ (x² – 9)e²ˣ dx

Answer 1

To evaluate the integral ∫ (x² – 9)e²ⁿ dx using integration by parts, we set u = x² – 9 and dv = e²ⁿ dx. By applying the integration by parts formula and simplifying the expression, the final result is determined to be (x²-9)(1/2)e²ⁿ - e²ⁿx + 9∫ e²ⁿ dx.

Step-by-step explanation:

To evaluate the integral ∫ (x² – 9)e²ⁿ dx using integration by parts, we can set u = x² – 9 and dv = e²ⁿ dx. Taking the derivative of u with respect to x gives du = 2x dx, and integrating dv with respect to x gives v = (1/2)e²ⁿ. Now we can use the integration by parts formula: ∫ u dv = uv - ∫ v du = (x²-9)(1/2)e²ⁿ - ∫ (1/2)e²ⁿ (2x dx).

Simplifying this expression gives us the final result of the integral: (x²-9)(1/2)e²ⁿ - e²ⁿx + 9∫ e²ⁿ dx.

As the integral of e²ⁿ dx can't be evaluated using elementary functions, this is the final result of the integral using integration by parts.