Question

Consider the integral ∫x⁵ eˣ dx. Applying the integration by parts technique, let u = __ and dv =__ dx. Then uv − ∫vdu = ___

Answer 1

In integration by parts for the integral of x⁵ e⁽x, let u = x⁵ and dv = e⁽x dx, then compute du and v to apply the formula uv - ∫ v du.

Step-by-step explanation:

The question asks how to apply integration by parts to the integral ∫x⁵ e⁽x dx. To use integration by parts, one typically lets u be a function that simplifies when differentiated, and dv be a function that simplifies when integrated. For this integral, a good choice would be to let u = x⁵ and dv = e⁽x dx. Then, differentiating u gives us du = 5x⁴ dx, and integrating dv gives v = e⁽x. Applying the integration by parts formula, ∫ u dv = uv - ∫ v du, we obtain x⁵ e⁽x - ∫5x⁴ e⁽x dx as the solution where uv - ∫vdu fits into the formula.