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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 x2exdx.

Letting f (x) = x2 and g(x) = ex, we have


x2exdx=x2ex|01 = 2xexdx  
 =e - 2xexdx  

Applying integration by parts again, we get


2xexdx=2xex|01 - 2exdx  
 =2e - (2ex|01)  
 =2  

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

x2exdx = 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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