How to Not Die in the Jaws of Calculus by Sullivan Thomas

Author:Sullivan, Thomas
Language: eng
Format: epub
Published: 2021-10-25T00:00:00+00:00

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∫ -∞∞f(x)dx = lim a→-∞limb→∞∫ abf(x)dx

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4.4 Substitution Rule

The substitution rule is much like the inverse chain rule and is a very helpful tool in solving more complicated integrals. It is defined by ∫ f(g(x))g′(x)dx = ∫ f(u)du. For example: If we wanted to solve I = ∫ dx, we couldn’t simply use ordinary integration techniques to do so. To solve it, we must first substitute u = x2 -1 du = 2xdx du∕2 = xdx I = ∫ () = ∫ du = u3∕2 +C and substituting u = x2 - 1 back again yields I = (x2 - 1)3∕2 + C.

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