Key takeaway: Sometimes we need to multiply or divide the entire integral by a constant, so we can achieve the appropriate form for u -substitution without changing the value of the integral.

In the u-substitution exercises for definite integrals, the next exercise asks us to integrate functions like 1/ (1+x²) and check our answer using the derivative of arctan (x), even though we haven’t learned …

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But it's a kind of a form of u substitution, where we do set u equal to something and see if it simplifies our expression in a way that, I guess, simplifies it.

U substitution is like the reverse chain rule. When we used the derivative chain rule, we saw that d/dx f (u (x)) = df (u)/du*du/dx. U substitution exploits this pattern in evaluating indefinite integrals.

Let me put a new color here just to ease the monotony. So it's going to be 2/21 times 7x plus 9 to the 3/2 power plus c. And we are done. We were able to take a kind of hairy looking integral and realize that …

For example, if it was a natural log of 0.5 or, who knows, whatever it might be. But then we are all done. We have simplified what seemed like a kind of daunting expression.

Does u-substitution apply, and if so how would we make that substitution? Well the key for u-substitution is to see, do I have some function and its derivative? And you might immediately recognize that the …

In these series of videos (U-substitution) you introduce the treatment of the derivative operators (dx, du, etc) as fractions. You specify that they really are not, but treat them like that anyway.