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Computing Integrals

Problems

Integration By Parts

Partial Fraction Decomposition

Problem : Compute x sin(x)dx .

Letting f (x) = x and g(x) = sin(x) in the formula for integration by parts, we have


x sin(x)dx = - x cos(x) - (- cos(x))dx  
  = - x cos(x) + sin(x) + C  

Problem : Find x 2 e x dx .

Letting f (x) = x 2 and g(x) = e x , we have


x 2 e x dx = x 2 e x|0 1 = 2xe x dx  
  = e - 2xe x dx  

Applying integration by parts again, we get


2xe x dx = 2xe x|0 1 - 2e x dx  
  = 2e - (2e x|0 1)  
  = 2  

Substituting the value of this integral back in the first computation yields

x 2 e x dx = e - 2    

Problem : Express sinn+2(x)dx in terms of sinn(x)dx by integrating by parts twice.

Let I = sinn+2(x)dx , the integral we are trying to compute. Integrating by parts gives


sinn+2(x)dx = sinn+1(x)sin(x)dx  
  = sinn+1(x)(- cos(x)) - (n + 1)sinn(x)cos(x)(- cos(x))dx  
  = -sinn+1(x)cos(x) + (n + 1)sinn(x)cos2(x)dx  

Integrating this new integral by parts, we have


sinn(x)cos2(x)dx = sinn(x)(1 - sin2(x))dx  
    = sinn(x)dx - sinn+2(x)dx  
    = sinn(x)dx - I  

Thus

I = - sinn+1(x)cos(x) + (n + 1) sinn(x)dx - I ,    

so

I = sinn+1(x)cos(x) + sinn(x)dx    

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