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In integral calculus, integration by reduction formulae is a method relying on recurrence relations. It is used when an expression containing an integer parameter , usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree , can't be integrated directly.
Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse A D ¯ {\displaystyle {\overline {AD}}} of the blue triangle has length 2 cos θ {\displaystyle 2\cos \theta } .
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List of trigonometric identities#Power-reduction formulae To a section : This is a redirect from a topic that does not have its own page to a section of a page on the subject. For redirects to embedded anchors on a page, use {{ R to anchor }} instead .
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator = (), and of the integration operator J {\displaystyle J} [ Note 1 ] J f ( x ) = ∫ 0 x f ( s ) d s , {\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}
With hindsight, however, it is considered the first general theorem of calculus to be discovered. [1] The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse ...
Perl's Math::BigInt module has a bmodpow() method to perform modular exponentiation; Raku has a built-in routine expmod. Go's big.Int type contains an Exp() (exponentiation) method whose third parameter, if non-nil, is the modulus; PHP's BC Math library has a bcpowmod() function to perform modular exponentiation
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an ...