Integral of e^√x / 2√x cos²(e^√x) - Watch 'USSR vs USA Integration by Substitution' first!

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  1. MathTheBeautiful
  2. 00) Chain Rule Review
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  4. x^3/sqrt(1-x^2) integral
  5. integral x^3 sqrt(1-x^2)
  6. how to integrate x^3/sqrt(1-x^2)
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My own approach is to identify the derivative which 'undoes' the integral so instead of asking what is the integral of this thing I am asking what would I have to differentiate to get this thing inside the integral sign. It's the exact same problem as taking the integral of course but now phrased in terms of an unknown differentiation. Here you can quickly spot it as related to the derivative of the tangent function and the argument of the tangent function is an exponential. So I write out what it looks like it has to be and simply take the derivative of that, call that the most likely trial function. So here that was writing down *_consider d/dx {tan[exp(x^1/2)]}_* which immediately reproduces the terms inside the integral sign. The only time I needed an integral sign at all was to write the answer out and at no stage am I switching variables in this way. I don't see the advantage in doing that once you see the integration problem submits to the chain rule, what am I missing?

muttleycrew