(*^ ::[paletteColors = 128; automaticGrouping; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = input, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakBelow, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12;] :[font = title; inactive; preserveAspect; startGroup; ] Techniques of Integration :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Integrals with Powers of Trig Functions :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Sines and Cosines :[font = input; preserveAspect; ] Integrate[Cos[x]^3 Sin[x]^2, x] :[font = input; preserveAspect; ] Expand[1/3 Sin[x]^3 - 1/5 Sin[x]^5, Trig -> True] :[font = text; inactive; preserveAspect; ] Need to load in Trig package for TrigReduce.. :[font = input; preserveAspect; ] < True] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup; ] Tans and Secs :[font = input; preserveAspect; ] Integrate[Sec[2x]^6, x] :[font = subsubsection; inactive; preserveAspect; startGroup; ] We see that Mathematica doesn't quite do what we'd like with Tangents and Secants in this problem. We can force it to do so by defining a rule (you are not expected to understand the details) to convert even powers of Sec to Tan: ;[s] 3:0,0;12,1;23,2;231,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,2,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = input; preserveAspect; ] sectotan = Sec[m_. x_]^n_Integer?Positive -> (1 - Tan[m x]^2)^Quotient[n,2] Sec[m x]^Mod[n,2]; :[font = input; preserveAspect; ] Integrate[Sec[2x]^6, x] /. sectotan // Expand :[font = input; preserveAspect; ] Integrate[Tan[x]^3 Sec[x]^3, x] :[font = input; preserveAspect; ] Integrate[Tan[x]^3 Sec[x]^3, x] /. sectotan // Expand :[font = input; preserveAspect; ] Integrate[Tan[x]^6, x] :[font = input; preserveAspect; endGroup; endGroup; endGroup; ] Integrate[Tan[x]^6, x] /. sectotan // Expand :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Sqrt[a^2 - x^2] Integrals :[font = input; preserveAspect; ] Integrate[1/Sqrt[a^2-x^2], x] :[font = input; preserveAspect; ] Integrate[1/Sqrt[a^2-x^2], x] //PowerExpand :[font = input; preserveAspect; ] Integrate[x^2/Sqrt[a^2-x^2], x] //PowerExpand :[font = input; preserveAspect; ] Integrate[x^4/Sqrt[a^2-x^2], x] //PowerExpand :[font = input; preserveAspect; endGroup; ] Integrate[x^6/Sqrt[a^2-x^2], x] //PowerExpand :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Hard Integral :[font = text; inactive; preserveAspect; ] Mathematica can not do the original integral: ;[s] 2:0,0;11,1;45,-1; 2:1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = input; preserveAspect; ] integrand1 = (Sqrt[1 - Cos[t]]) /Cos[t]; Integrate[integrand1, t] :[font = text; inactive; preserveAspect; ] Carry out the substitution u = cos t: :[font = input; preserveAspect; ] Integrate[-1/(u Sqrt[1 + u]), u] :[font = text; inactive; preserveAspect; ] v = sqrt(1 + u) : :[font = input; preserveAspect; ] Integrate[2/(1 - v^2), v] :[font = input; preserveAspect; endGroup; ] % /. v -> Sqrt[1 + u] /. u -> Cos[t] :[font = section; inactive; Cclosed; preserveAspect; startGroup; ] Method of Substitution in Mathematica :[font = text; inactive; preserveAspect; ] This is just for the Instructors. We know that the differential is a formal guide to change of variables in an integral: f(x) dx becomes f(g(u)) g'(u) du under the one-to-one change of variables x=g(u). The operator \verb:Dt[]: can be used to represent the differential in \Mma: :[font = input; preserveAspect; ] Dt[f[x]] :[font = text; inactive; preserveAspect; ] Illustrative Example: Simplify the integral of 1/( 1+e^(2x) ) by making the change of variables u = e^x. To apply the \verb:Dt[]: operator, we need the substitution in the form x = g(u) (inverse function!), that is, x = ln u: :[font = input; preserveAspect; ] Dt[x]/Sqrt[1 + E^(2x)] /. x -> Log[u] :[font = input; preserveAspect; ] integrand1 = (Sqrt[1 - Cos[t]])/Cos[t] Dt[t] /. t -> ArcCos[u] // Simplify :[font = text; inactive; preserveAspect; ] Forcing Mma to dump that annoying factor: :[font = input; preserveAspect; ] integrand1 = Sqrt[(integrand1/Dt[u])^2] Dt[u] //PowerExpand :[font = input; preserveAspect; endGroup; endGroup; ] integrand2 = integrand1 /. u -> v^2 - 1 //PowerExpand ^*)