Lesson 13: Integration Techniques Summary and ReviewA Substitution Problem:An Integration by Parts Problem:A Trig Substitution:A Partial Fractions ProblemExercises:Calculus IILesson 13: Integration Techniques Summary and Review You can use the student package in Maple to practice your integration techniques. First load the student package by typing> with(student):Then read over the help screens on changevar, intparts, and value, paying particular attention to the examples at the bottom of the screens. Here are some examples. A Substitution Problem: Integration by substitution is based on the chain rule. Thus if we have an integral which looks like , then by make the change of variable, and letting we have a new, perhaps simpler integral, to work on. In Maple this is accomplished using the word changevar from the student package.Find an antiderivative of > F := Int(1/sqrt(1+sqrt(x)),x);Let's try the change of variable sqrt(x) = u .> G := changevar(sqrt(x)=u,F);This does not seem to help. Lets try 1 + sqrt(x)= u> G := changevar(1+sqrt(x)=u,F);Now we can do it by inspection, so just finish it off.> G := value(G);Now substitute back and add in the constant.> F := subs(u=sqrt(x),G) + C;Integration by substitution is the method use try after you decide you can't find the antiderivative by inspection.An Integration by Parts Problem: Integration by parts is based on the product rule for derivatives. It is usually written. It turns one integration problem into one which 'may' be more doable. Once you decide to use parts, the problem is what part of the integrand to let be u. Integrate > F := Int(x^2*arctan(x),x);The word is intparts. Let's try letting .> G := intparts(F,x^2);That was a bad choice. Try letting > G := intparts(F,arctan(x));This is much more promising. Split off the integral on the end.> H := op(2,G);Now do a partial fractions decomposition of the integrand of H, using parfrac. > H:= Int(convert(integrand(H),parfrac,x),x); Now we can do it by inspection. > H1 := 1/6*x^2 - 1/3*1/2*ln(1+x^2);Let's check this with the student value. > simplify(value(H-H1));Note the difference of a constant, which is fine for antiderivatives.ETAIL: The problem of choosing which part of the integrand to assign to u can often besolved quickly by following the etail convention. If your integrand has an Exponential factor, choose that for u, otherwise if it has a Trigonometric factor, let that be u, otherwisechoose an Algebraic factor for u, otherwise chose an Inverse trig function, and as a last resort choose u to be a logarithmic factor. Let dv be what's left over. A Trig Substitution:Find an antiderivative of> F := Int(x^3/sqrt(x^2+1),x);The presence of suggests letting .> G := changevar(x=tan(t),F,t);Now use the trig identity .> G := subs(sqrt(1+tan(t)^2)=sec(t),G);Another substitution into the integrand.> G := subs(tan(t)^3 = (sec(t)^2-1)*tan(t),G);Let's make a change of variable,> H := changevar(sec(t)=u,G);From here, we can do it by inspection.> H := value(H);Now unwind the substitutions.> G := subs(u=sec(t),H);> F := subs(t = arctan(x),G);> F := subs(sec(arctan(x))=sqrt(1+x^2),F) + C;Checking this calculation:> F1 := int(x^3/sqrt(x^2+1),x);It looks different, but is it?> simplify(F-F1);Yes, but only by a constant.A Partial Fractions Problem Integrate the rational function> y :=(4*x^2+x -1 )/(x^2*(x-1)*(x^2+1));First get the partial fractions decomposition of y.> y := convert(y,parfrac,x);We can almost do this by inspection, except for the last term.> F := Int(y,x); > F := expand(F);Now we can do each one by inspection. So we'll just use value .> F := value(F) + C;Exercises:Exercise: Use the student package to perform the following integrations. Exercise: Find the area of the region enclosed by the x-axis and the curve on the interval . Sketch the region. Then find the vertical line that divides the region in half and plot it.Exercise: Find the length of the graph of the parabola from O(0,0) to P(10,100). Find the point on the graph which is 10 units from O along the graph. Make a sketch, showing the points O, P, and Q on the graph.Exercise: Find the volume of the solid of revolution obtained by revolving the region trapped between the the graph of on and the x-axis about the x-axis. Sketch a graph. Does this volume approach a finite limit as n gets