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Computing Integrals

Problems

Change of Variables

Integration By Parts

Problem : Compute sin2(x)cos(x)dx .

Making the change of variables u = sin(x) , the integral becomes

u 2 du = + C    

Substituting the definition of u , we obtain (sin3(x))/3 + C . It is easy to check that this has derivative equal to sin2(x)cos(x) .

Problem : Find 2xe x2+1 dx .

Letting u = 1 + x 2 , we have

2xe x2+1 dx = e u du = e u|1 2 = e 2 - e.    

Problem : Compute dx .

Setting u = log(x) , we have

dx = du = log(u) + C = log(log(x)) + C    

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