Math 1A: introduction to functions and calculus Oliver Knill, 2011Lecture 28: SubstitutionIf we differentiate the function sin(x2) and use the chain rule, we get cos(x2)2x. By the fundamentaltheorem of calculus, the anti derivative of cos(x2)2x is sin(x2). We know thereforeZcos(x2)2x dx = sin(x2) + C .Spotting the chain ruleHow can we see the integral without knowing the result already? Here is a very important case:If we can spot that f(x) = g(u(x))u′(x), then t he anti derivative of f is G(u(x))where G is the anti derivative of g.1 Find the a nti derivative ofex4+x2(4x3+ 2x) .Solution: The derivative of the inner function is to the rig ht.2 FindZ√x5+ 1x4dx .Solution. The derivative of x5+ 1 is 5x4. This is almost what we have there but theconstant can be adapted. The answer is (1/5)(x5+ 1)3/2.3 Find the a nti derivative oflog(x)x.Solution: The derivative of log(x) is 1/x. The antiderivative is log(x)2/2.4 Find the a nti derivative ofcos(sin(x2)) cos(x)2x .Solution. We see the derivative of sin(x2) appear o n the right. Therefore, we havesin(sin(x2)).In the next three examples, substitution is actually not necessary. You can just writedown the anti derivative, and adjust the constant. It uses the following ”speedy rule”:IfRf(ax + b) dx = F (ax + b)/a where F is the anti derivative of f.125R√x + 1 dx. Solution: (x + 1)3/2(2/3).6R11+(5x+2)2dx. Solution: arctan(5 x + 2) (1/5).Doing substitutionSpotting thing s is sometimes not easy. The method of substitution helps to formalize this. To doso, identify a part of the formula to integrate and call it u then replace an occurrence of u′dx withdu.Zf(u(x) ) u’(x) dx =Zf( u ) du .Here is a more detailed description: replace a prominent part o f the function with a new variableu, then use du = u′(x)dx to replace dx with du/u′. We aim to end up with an integralRg(u) duwhich does not involve x anymore. Finally, after integration of this integral, replace the va r iableu again with the function u(x). The last step is called back-substitution.7 Find the a nti-derivativeZlog(log(x))/x dx .Solution Replace log(x) with u and replace u′dx = 1/xdx with du. This givesRlog(u) du =u log(u) − u = log(x) log log(x)) − log(x).8 Solve the integralZx/(1 + x4) dx .Solution Substitute u = x2, du = 2xdx to get (1/2)Rdu/(1 + u2) du = (1/2) arctan(u) =(1/2) arctan(x2).9 Solve the integralZsin(√x)/√x .Here are some examples which are not so straightforward:10 Solve the integralZsin3(x) dx .3Solution. We replace sin2(x) with 1 − cos2(x) to getZsin3(x) dx =Zsin(x)(1 − cos2(x)) dx = −cos(x) + cos3(x)/3 .11 Solve the integralZx2+ 1√x + 1dx .Solution: Substitute u =√x + 1. This gives x = u2− 1, dx = 2udu and we getR2(u2−1)2+ 1 du.12 Solve the integralZx3√x2+ 1dx .Trying u =√x2+ 1 but this does not work. Try u = x2+ 1, then du = 2xdx anddx = du/(2√u − 1). Substitute this in to getZ√u − 132√u − 1√udu =Z(u − 1)2√u=Zu1/2/2−u−1/2/2 du = u3/2/3−u1/2=(x2+ 1)3/23−(x2+1)1/2.Definite integralsWhen doing definite integrals, we could find the antiderivative as described and then fill in theboundary points. Substituting the boundaries directly accelerates the process since we do nothave to substitute back to the original variables:Zbag(u(x))u′(x) dx =Zu(b)u(a)g(u) du .Pro of. This identity follows from the fact that the right hand side is G(u(b)) − G(u(a)) by thefundamental theorem of calculus. The integrand on the left has the anti derivative G(u(x)). Againby the fundamental theorem of calculus the integral leads to G(u(b)) − G(u(a)).Top: To keep track which bounds we consider it can help to writeRx=bx=af(x) dx.13 Find the anti derivative ofR20sin(x3− 1)x2dx. Solution.Zx=2x=0sin(x3+ 1)x2dx .Solution: Use u = x3+ 1 and get du = 3x2dx. We getZu=7u=1sin(u)du/3 = (1/3) cos(u)|71= [−cos(7) + cos(1)]/3 .Also here, we can see the integrals directlyTo integrate f(Ax + B) from a to b we get [F (Ab + B) −F (Aa + B)]/A, where Fis the anti-derivative of f.14R1015x+1dx = [log(u)]/5|61= log(6)/5.15R53exp(4x − 10) dx = [exp(10) − exp(2)]/4.4Homework1 Find the f ollowing anti derivatives.a)R5x sin(x2) dxb)Rex5+x(x4+ 1/5) dxc) cos(cos(x)) sin(x)d) etan(x)/ cos2(x).2 Find the f ollowing definite integrals.a)R53√x5+ x(x4+ 1/5) dxb)R√π0sin(x2)x dx.c)Re1/e√log(x)xdx.d)R10x/√1 + x2dx.3 a) Find the integralR103x√1 − x4dx using a substitution and interpreting the resultusing a known area.b) Find the moment of inertia of a rod with density f(x) =√x3+ 1 between x = 0and x = 4.4 a) IntegrateZ10arcsin(x)√1 − x2dx .b) Find the definite integralZ6eedxplog(x)x.5 a) Find the indefinite integralZx5√x2+ 1dx .b) Find the anti-derivative off(x) =1x(1 +
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