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

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

u2du = + 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 2xex2+1dx.

Letting u = 1 + x2, we have

2xex2+1dx = eudu = eu|12 = e2 - e.    

Problem : Compute dx.

Setting u = log(x), we have

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

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