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Introduction to Integrals

Problems for "Methods of Calculating Integrals"

Methods of Calculating Integrals

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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    

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