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Problems for "Methods of Calculating Integrals"

Problems for "Methods of Calculating Integrals"

Problems for "Methods of Calculating Integrals"

Problems for "Methods of Calculating Integrals"

Problems for "Methods of Calculating Integrals"

Problems for "Methods of Calculating Integrals"

Problem : Evaluate x(2x 2+4)3 dx .


u   = 2x 2 +4  
  = 4x  
dx   =  

Now substitute back into the expression:


x u 3   = u 3 du =  
    = 2x 2+4 + c  

Problem : Evaluate 4x cos(x 2+2)dx.


u   = x 2 +2  
  = 2x  
dx   =  
4x cos(u)   = 2cos(u)du = 2 sin(u)  
    = 2 sin(x 2 + 2) + c  

Problem : Evaluate (x )dx.


u   = x 2 -5  
  = 2x  
dx   =  
x   = du = = u  
    = x 2-5 + c  

Problem : Evaluate (3x-4)2 dx.


    u = 3x - 4  
    = 3  
    dx = du  
    u 2 du  

(Note: the limits of integration have been removed, since they do not apply to u but to x .

u 2 du = u 3    

Now substitute x back into the result.


    = 3x-4 0 1  
    = 3(1)-4-(-4)3  
    = (-1)-(-64)  
    = = 7  

Problem : Use the trapezoid rule with five subdivisions to approximate the area under f (x) = x 2 + 1 on [1, 6] .


Area   2+2(5)+2(10+2(17)+2(26)+37(1)  
    = 77.5  

This compares well with

16x2+1 dx=75