Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.08 Office Hours 9:15-10:15 MTWThF Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. Friday, February 8 7.2 Integration by Parts p391: 32,42,46,50 u v Main Idea: du dv n n n x Key Words: Integration by parts. x Sin[x], x Cos[x], x e, n x x ln(x), x ln(x), e Sin[x], e Cos[x] Goal: Learn how to integrate the above expressions. -------------------------------------------------------- Previous Assignment: p386: (26),36),(42) Page 386 Problem 26 x=Pi/6 Cos[x] INT 2 Sin[x] dx x=0 u = Cos[x] du = -Sin[x] x=Pi/6 u INT - 2 du x=0 _ _ x=Pi/6 | u | u=Sqrt[3]/2 | -2 | -1 Sqrt[3]/2 | ----- | = ----- ( 2 - 2 ) = 0.255884 |_ ln(2) _| x=0 ln(2) u=1 ---------------------------------------- Get["font.math"]; f[x_] := 2^Cos[x] Sin[x]; Integrate[f[x],{x,0, Pi/6}] P1 = Plot[f[x],{x,-2 Pi,0}]; P2 = Plot[f[x],{x, 0, Pi/6},PlotStyle->{RGBColor[1,0,0]}]; P3 = Plot[f[x],{x,Pi/6, Pi}]; P4 = ParametricPlot[{Pi/6,y},{y,0,f[Pi/6]},PlotStyle->{RGBColor[1,0,0]} ]; P5 = Show[P1,P2,P3,P4,PlotLabel->"P386 P26: f(x) = 2^Cos[x] Sin[x]", PlotRange->All,AspectRatio->Automatic]; Display["p26.ps",P5]; ---------------------------------------------------------------- Page 386 Problem 36 2 Csc [2 t] INT ----------------- dt Sqrt[ 1+Cot[2t] ] u = 1+Ctn[2t] 2 du = -Csc [2t] 2 dt 1/2 - 1/2 du u INT ----------- = -1/2 ----- + C = - Sqrt[1+Cot[2t] ] Sqrt[u] 1/2 / -1/2 2 \/ -(1/2) Sqrt[1+Ctn[2t]] (- Csc 2t] 2 Integrate[Csc[2t]^2/Sqrt[1+Cot[2t]],t] ------------------------------------------------------- Page 386 Problem 42 5 INT ------------- dx 2 Sqrt[ 9-4x ] 5 INT ------------------- dx 2 3 Sqrt[ 9/9-4x /9 ] dx 5/3 INT ----------------- Sqrt[1-(2x/3)^2] 2/3 dx 5/3 3/2 INT ----------------- Sqrt[1-(2x/3)^2] 5/2 ArcSin[2x/3] + C / 2/3 5 \/ 5/2 -------------- = -------------- Sqrt[1-(2x/3)^2] Sqrt[9-4x^2] Integrate[5/Sqrt[9-4x^2],x] ====================================================== New Material Page 387 Example 1 INT x Cos[x] dx x Sin[x] dx Cos[x] dx = x Sin[x] - INT Sin[x] dx = x Sin[x] + Cos[x] + C / \/ Sin[x] + x Cos[x] - Sin[x] = x Cos[x]. Theory: There are several ways to present the material. The slicker it is, the less it should be trusted. By impressive mathematics: d(uv) = u dv + v du so INT u dv = uv - v du By direct approach: INT x Cos[x] has to have something to give x Cos[x] so let's put x Sin[x]. But then the derivative does not check since / (x Sin[x]) = Sin[x] + x Cos[x]. So we simply throw in another term whose derivative cancels the Sin[x]. Namely Cos[x]. The final answer is x Sin[x] + Cos[x] + C =============================================================== To keep things straight arrange the problem in a 2x2 array. u v du dv The top row has the UN-differentiated functions. The bottom row has the derivatives. The original problem is on the main diagonal. The answer is the top uv minus the other diagonal. ========================================================== x=2 Page 388 Example 2. INT ln(x) dx x=1 ln(x) x dx/x dx = x ln(x) - INT dx | | x=2 = | x ln(x) - x | 2 ln(2)-2 -(0-1) = 2 ln(2) -1 = 0.386294 <===Answer | | x=1 / \/ ln(x) + x 1/x - 1 = ln(x) ----------------------------------------------------- Get["font.math"]; Integrate[Log[x],{x,1,2}] f[x_] := Log[x]; P1 = Plot[f[x],{x,1/2,1}]; P2 = Plot[f[x],{x,1,2},PlotStyle->{RGBColor[1,0,0]}]; P3 = Plot[f[x],{x,2,3}]; P4 = ParametricPlot[{2,y},{y,0,f[2]},PlotStyle->{RGBColor[1,0,0]}]; P5 = ParametricPlot[{0,y},{y,-1,1}]; P6 = Show[P1,P2,P3,P4,P5,PlotLabel->"P388 Ex 2: INT ln(x) on (1,2)"]; Display["ex2.ps",P6]; ====================================================== Page 388 Example 3. INT ArcSin[x] dx ArcSin[x] x dx ----------- dx Sqrt[1-x^2] x dx = x ArcSin[x] - INT ----------- Sqrt[1-x^2] = x ArcSin[x] + Sqrt[1-x^2] + C <== answer / x \/ ArcSin[x] + ------------ +1/2 (1-x^2)^(-1/2) (-2x) Sqrt[1-x^2] =============================================================== Page 389 Example 5 2 INT x Sin[x] dx 2 ans: -x Cos[x] + 2 x Sin[x] + 2 Cos[x] + C Page 389 Example 6 x INT e Sin[x] dx x ans: 1/2 e (Sin[x]-Cos[x]) + C =========================================================== n Find a reduction formula for INT Sin[x] dx n-1 -Sin [x] Cos[x] n-1 n-2 ans: -------------------- + ------ INT Sin x dx n n ============================================================ x=Pi/2 8 INT Sin [x] dx x=0 ans: 7/8 5/6 3/4 1/2 Pi/2 = 35/256 Pi ======================================================== In class problems x INT x e dx INT t Sqrt[t+1] INT ArcTan[x] dx n Find a reduction formula for INT Sec [x] dx