# Lecture 6 The Substitution Method - W15 MAT 21B LECTURE 6...

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W15 MAT 21B LECTURE 6: THE SUBSTITUTION METHODGEORGE MOSSESSIANUp til now, we have defined the definite integralRbaf(x) as the limit, with the number of rectangles goingtoof the Riemann sum which estimates the area underf(x) over the interval [a, b], and then we demon-strated the Fundamental Theorem of Calculus, which allows us to computeRbaf(x) asF(b)-F(a), whereF(x) is the antiderivative off(x). Notice that the constant term of the antiderivative doesn’t matter, sinceit would get cancelled in the subtraction. However, theindefiniteintegralRf(x)dxis just defined asF(x)+C.Sometimes, it’s very obvious how to take the antiderivative, for exampleZ0-π/4secxtanxdx= sec(0)-sec(-π/4) = 1-2,orZ4132x-4x2dx=x3/2+2x41=112.The notationF(x)bais shorthand forF(b)-F(a). It comes from the convention thatdfdx(a) can also bewritten asddxf(x)a, which is a useful notation becauseddxf(a) = 0, becausef(a) is just a constant.EXAMPLESometimes, it’s not quite so obvious, for example, inR(x3+x)5(3x2+ 1)dx.Now, youcertainly could use binomial coefficients and expand (x3+x)5, multiply it by the other term, get a polynomial,and integrate that term by term. But it’smuch