Integrate x^3 * lnx
I would start by asking the student the different forms of integration we know (e.g. substitution, parts).I would then ask the student whether parts or substitution gives the easiest method.If the student isn't sure which method to use, I would ask them to try substitution first so that they can see that substitution doesn't lead to a simplification of the problem.
Integration by parts formula: ∫ u(dv/dx) dx = uv − ∫ v(du/dx) dxIf the student is unsure about which term (x3 and lnx) to use for u and v, I would ask them to consider which term is difficult to integrate and easy to differentiate. This should lead them to u = lnx and dv/dx = x3.
u = lnx dv/dx = x3 du/dx = 1/x v = x4/4
Then simply substitute the values u, v, du/dx, dv/dx into the parts formula to solve for the answer.
(x4/4lnx) - 1/4∫x3 = (x4/4lnx) - x4/16 + C
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