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In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution .
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Windows 10: ⊞ Win+x > u > s. Windows 7: ⊞ Win+→+→+↵ Enter. Sleep (available on some keyboards) ⌥ Opt+⌘ Cmd+Eject: Sleep (available on some keyboards, configurable in Control Panel Power Options Advanced tab dialog box) Shut down computer Windows 10: ⊞ Win+x > u > u: Ctrl+⌥ Opt+⌘ Cmd+Eject (no confirmation, shutdown is immediate)
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Integration by parts is a heuristic rather than a purely mechanical process for solving integrals; given a single function to integrate, the typical strategy is to carefully separate this single function into a product of two functions u(x)v(x) such that the residual integral from the integration by parts formula is easier to evaluate than the ...
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