Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Spring1B/Find two or three classmates and a few feet of chalkboard. Introduce yourself to your newfriends, and write all of your names at the top of the chalkboard. As a group, try your hand atthe following exercises. Be sure to discuss how to solve the exercises — how you get the solutionis much more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: some are very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, or are independently marked.Always draw pictures.Integration by PartsIn 1A, you learned the product rule for differentiation:ddx[f(x) g(x)] =ddx[f(x)] g(x) + f(x)ddx[g(x)]This is often abbreviated (fg)0= f0g + g0f, or d(fg) = f dg + g df. By the fundamental theoremof calculus, the integral of the derivative of a function is the original function. Thus fg + C =Rd(fg) =R[f dg + g df] =Rf dg +Rg df. Rearranging the equality gives the integration by partsformula, which is a kind of product rule for indefinite integrals:Zf(x) g0(x) dx = f(x) g(x) −Zg(x) f0(x) dxWhat about definite integrals? ThenRbaf dg +Rbag df =Rbad(fg) = [fg]ba= f(b)g(b) − f(a)g(a):Zbaf(x) g0(x) dx = [f(x) g(x)]ba−Zbag(x) f0(x) dx1. (a) Use the integration by parts formula to integrateZx exdxHint: let f(x) = x and g0(x) = ex. What is g(x)?(b) Based on your answer to part (a), findRx2exdx. Based on that, findRx3exdx andRx4exdx.(c) Guess the pattern from part (b). Prove your pattern: use integration-by-parts to writeRxnexdx in terms ofRxn−1exdx. This is an example of a reduction formula.2. (a) Use the integration by parts formula to integrateZln x dxHint: let f(x) = ln x and g0(x) = 1.1(b) Use the integration by parts formula to integrateZ(ln x)2dxHint: let f(x) = (ln x)2and g0(x) = 1.(c) Use the integration by parts formula to integrateZ(ln x)ndx3. Find a formula forRf(x) g00(x) dx by applying the integration-by-parts formula twice. Sup-pose that f(1) = 2, f(4) = 7, f0(1) = 5, and f0(4) = 3 (and that f00(x) is continuous on[1, 4]). What isR41x f00(x) dx?4. By integrating by parts twice and rearranging, findRexcos x dx. What isRexsin x dx? HowaboutReaxcos x dx, where a is a constant? How aboutRx excos x dx?5. § Use integration by parts to evaluate the following integrals. For some you will first need tomake a substitution.(a)Zt sin 2t dt (b)Zx2sin πx dx (c)Zarcsin x dx(d)Zs 2sds (e)Z10(x2+ 1)e−xdx (f)Z94ln y√ydy(g)Zπ0x3cos x dx (h)Z√31arctan(1/x) dx (i)Zt0essin(t − s) ds(j)Z√π√π/2θ3cos(θ2) dθ (k)Zπ0ecos tsin 2t dt (l)Zsin(ln x) dx6. What’s wrong with the following proof that 0 = 1?ln x =Z1xdx =1xx −Z−1x2x dx = 1 +Z1xdx = 1 + ln x7. Why can we forget to add an arbitrary constant during the intermediate steps when integrat-ing by parts? IntegrateRxnexdx completely honestly: let u = xnand dv = exdx, but thistime let v = ex+ C.8. Let n = 2k + 1 be an odd integer. CalculateRπ/2x=0cosnx dx in two different ways:(a) Using a reduction formula. What happens to the boundary terms (the uv inRu dv =uv −Rv du)? Why does it matter that n is odd? (The formula is different for even n.)(b) Using a u-substitution. (Hint: cos2x = 1 − sin2x, and sin0x = cos x.) For any givenk, you could then expand out and evaluate the integral. For a general k, you can useintegration by parts to get a reduction formula.9. What isRπ/20dx? Use the reduction formula from the previous problem to evaluateRπ/20cosnx dxfor n
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