What is the indefinite integral of ln(x) ?
We can use integration by parts to solve this question. If we look at the formula for Integration by parts: ∫u(dv/dx)dx = uv - ∫ v (du/dx) dx, we see that u must be multiplied by something else so therefore, when we make u = ln(x), we put (dv/dx) = 1 (This is because ln(x)1 is still ln(x)). So in order to get v we integrate 1 with respect to x, and we get x. So, u = ln(x), v = x, (du/dx) = 1/x, (dv/dx) = 1.
And therefore, substituting everything into the formula , we get: ln(x) * x - ∫x(1/x) dx. It follows through that ∫x*(1/x) dx becomes ∫1 dx which integrates to x. Putting all the parts together gives: xln(x)-x. We must also remember the constant of integration, and so, the final answer becomes: xln(x)-x+C.
DM
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