integration_by_parts - INTEGRATION BY PARTS Integration by...

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INTEGRATION BY PARTSIntegration by parts is essentially the reverse of product rule.ddx[f(x)g(x)] =f0(x)g(x) +g0(x)f(x)f(x)g(x) =Zf0(x)g(x)dx+Zg0(x)f(x)dxZf(x)g0(x)dx=f(x)g(x)-Zf0(x)g(x)dxAn easier way to remember this is usually the following:Theorem 1.Ru dv=uv-Rv duIn practice, we will be chooseuanddv. The following is a basic example:Example 1.FindRxexdx.Solution:We will need to chooseuanddv. We chooseu=xanddv=exdx,hencedu=dxandv=ex. Hence, applying the formula:Zxexdx=xex-Zexdx=xex-ex+CThere is sort of a general rule for choosingu[it works almost all the time] andthe rule is ’ILATE’. To explain the rule, let’s start with what each letter therestands for:Inverse Trigonometric functionsLogarithm functionsAlgebraic expressionsTrigonometric functionsExponential functionsWe chooseuaccording to this rule where the leftmost is given the preference.(Example: choose inverse trig. over exponentials)Here is an example to illustrate the rule:

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