# Lecture2-slides-substitution.pdf - Lecture 2 Warm-Up...

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Lecture 2Warm-Up ProblemsSuppose thatf(x) =Z20xp1-5u3du.Findf0(x).Find the following indefinite integrals(Hint: simplify using the distributive property)1.Zsec2(x)(1 + 2 cos2(x))dx2.Z(x2+ 3)22x dx3.Z(x2+ 3)1002x dxAnnouncements:IQuiz 1 on Tuesday, Sept 11: FTC and SubstitutionIHomework 0 (optional) and Homework 1 are posted.HW0 is due on Sept 12, HW1 is due on Sept 13, both at 8:45am.
Recall from last classTheorem (The Fundamental Theorem of Calculus)Suppose that f(x)is continuous on[a,b]. Then,1.if g(x) =Rxaf(t)dt for all x[a,b], then g0(x) =f(x). In other words,ddxZxaf(t)dt=f(x).2.Suppose F(x)is an antiderivative of f(x)(i.e. F0(x) =f(x)), thenZbaf(x)dx=F(b)-F(a).
Recall from last classTheorem (The Fundamental Theorem of Calculus)Suppose that f(x)is continuous on[a,b]. Then,1.if g(x) =Rxaf(t)dt for all x[a,b], then g0(x) =f(x). In other words,ddxZxaf(t)dt=f(x).2.Suppose F(x)is an antiderivative of f(x)(i.e. F0(x) =f(x)), thenZbaf(x)dx=F(b)-F(a).“Integrals undo derivatives”
Recall from last classTheorem (The Fundamental Theorem of Calculus)Suppose that f(x)is continuous on[a,b]. Then,1.if g(x) =Rxaf(t)dt for all x[a,b], then g0(x) =f(x). In other words,ddxZxaf(t)dt=f(x).2.Suppose F(x)is an antiderivative of f(x)(i.e. F0(x) =f(x)), thenZbaf(x)dx=F(b)-F(a).“Integrals undo derivatives”Properties of Definite Integrals1.Zbaf(x)dx=-Zabf(x)dx
Recall from last classTheorem (The Fundamental Theorem of Calculus)Suppose that f(x)is continuous on[a,b]. Then,1.if g(x) =Rxaf(t)dt for all x[a,b], then g0(x) =f(x). In other words,ddxZxaf(t)dt=f(x).2.Suppose F(x)is an antiderivative of f(x)(i.e. F0(x) =f(x)), thenZbaf(x)dx=F(b)-F(a).“Integrals undo derivatives”Properties of Definite Integrals1.Zbaf(x)dx=-Zabf(x)dx2.Zbaf(x)dx=Zcaf(x)dx+Zbcf(x)dx
ExampleWorksheet 1, Problem 3b:Find the derivative ofh(z) =Z2ztan1u2+ 1du
ExampleWorksheet 1, Problem 3b:Find the derivative ofh(z) =Z2ztan1u2+ 1duh0(z) =ddzZ2ztan1u2+ 1du=ddz-Zz2tan1u2+ 1du=-tan1z2+ 1
FTC (part 1) + Chain RuleWorksheet 1, Problem 3c:Find the derivative ofk(x) =Zx-2xsec2(s)ds
FTC (part 1) + Chain RuleWorksheet 1, Problem 3c:Find the derivative ofk(x) =Zx-2xsec2(s)dsk(x) =Z0-2xsec2(s)ds+Zx0sec2(s)ds
FTC (part 1) + Chain RuleWorksheet 1, Problem 3c:Find the derivative ofk(x) =Zx-2xsec2(s)dsk(x) =Z0-2xsec2(s)ds+Zx0sec2(s)ds=-Z-2x0sec2(s)ds+Zx0sec2(s)ds
FTC (part 1) + Chain RuleWorksheet 1, Problem 3c:Find the derivative ofk(x) =Zx-2xsec2(s)dsk(x) =Z0-2xsec2(s)ds+Zx0sec2(s)ds=-Z-2x0sec2(s)ds+Zx0sec2(s)dsk0(x) =-sec2(-2x)ddx[-2x] + sec2(x)
FTC (part 1) + Chain RuleWorksheet 1, Problem 3c:Find the derivative ofk(x) =Zx-2xsec2(s)dsk(x) =Z0-2xsec2(s)ds+Zx0sec2(s)ds=-Z-2x0sec2(s)ds+Zx0sec2(s)dsk0(x) =-sec2(-2x)ddx[-2x] + sec2(x)= 2 sec2(-2x) + sec2(x)
FTC (part 1) + Chain Ruleddx"Zh(x)af(t)dt#=f(h(x))h0(x)
FTC (part 1) + Chain RuleExample:Find the derivative of(θ) =Zsin(θ)21t3-tdt.

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