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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."
Substitution (trigonometric, tangent half-angle, Euler) Euler's formula; Partial fractions (Heaviside's method) Changing order; Reduction formulae; Differentiating under the integral sign; Risch algorithm
Euler substitution is a method for evaluating integrals of the form ... As we can see this is a rational function which can be solved using partial fractions.
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 .
because of the substitution rule for integrals. If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat the derivative as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in ...
Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial:
Substitution (trigonometric, tangent half-angle, Euler) Euler's formula; Partial fractions (Heaviside's method) Changing order ... A similar method can be used to ...
Inverse chain rule method; Integration by substitution. Tangent half-angle substitution; Differentiation under the integral sign; Trigonometric substitution; Partial fractions in integration. Quadratic integral; Proof that 22/7 exceeds π; Trapezium rule; Integral of the secant function; Integral of secant cubed; Arclength; Solid of revolution ...
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