Sections5-2andthefirsthalfofSection5-3Finished-SeeAlsoApril10Finished

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Section 5.2 — Integration by SubstitutionExampleComputeZ(5x+ 7)9dxHow can we go about evaluating this integral?One option would be tomultiply it out, and integrate the resulting polynomial, but this seems likeit would be a rather daunting choice. What other options might we have?Well, if we were differentiating something that looked like these, we wouldthink of the integrand asu9whereu(x) = 5x+ 7, and then we would use thechain rule on the function. Let’s try doing something similar here. Takeu= 5x+ 7 then, differentiatingdudx= 5du= 5dxdx=15duThis makes our integralZu915duWhich we can integrate using our rules from the previous section.Using Substitution to IntegrateRf(x)dx(1) Choose a substitutionu=u(x) that ”simplifies” the integrandf(x).(2) Express the entire integral in terms ofuanddu=u0(x)dx. This means that all termsinvolvingxanddxmust be transformed to terms involvinguanddu.(3) When step (2) is complete, the given integral should have the formZf(x)dx=Zg(u)duIf possible, evaluate this transformed integral by finding an antiderivativeG(u) forg(u).(4) Replaceubyu(x) inG(u) to obtain an antiderivativeG(u(x)) forf(x) so thatZf(x)dx

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