What is the antiderivative of #sec^2(x)#?
Apr 03, · The antiderivative of sec (x) is equal to ln |sec (x) + tan (x)| + C, where C represents a constant. This antiderivative, also known as an integral, can be solved by using the integration technique known as substitution. The antiderivative of sec (x) is equal to the antiderivative of sec (x) * ([sec (x) + tan (x])/ [sec (x) + tan (x)]). The antiderivative of sec(x) is ln |secx + tanx| + C, where C is a constant. When faced with ∫ sec(x) dx, it is definitely not obvious where we See full answer below.
The following is a list of integrals antiderivative functions of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.
In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration. An integral that is what is the antiderivative of sec rational function of the sine and cosine can be evaluated using Bioche's rules.
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This is a summary of differentiation rules , that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are functions of real numbers R that return real values; although more generally, the formulae below apply wherever they are well defined   — including the case of complex numbers C.
In Leibniz's notation this is written as:. Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial. The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule. If f and g are functions, then:. The elementary power rule generalizes considerably. The most general power rule is the functional power rule : for any functions f and g ,. The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function using the chain rule :.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for taking derivatives.
For the first and fourth quadrant i. Its partial derivatives are. See Hyperbolic functions for restrictions on these derivatives. This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. Some rules exist for computing the n - th derivative of functions, where n is a positive integer. These include:. If f and g are n -times differentiable, then. These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics.
Those in this article in addition to the above references can be found in:. From Wikipedia, the free encyclopedia. Redirected from List of differentiation identities. Wikimedia list article with rules for computing the derivative of a function in calculus.
Limits of functions Continuity. Mean value theorem Rolle's theorem. Definitions Derivative generalizations Differential infinitesimal of a function total. Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem. Lists of integrals Integral transform Definitions Antiderivative Integral improper Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution trigonometric , Weierstrass , Euler Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm.
Fractional Malliavin Stochastic Variations. Main article: Product rule. Main article: Chain rule. Main article: Inverse functions and differentiation. Main article: Power rule. Main article: Reciprocal rule. Main article: Quotient rule. Main article: Differentiation of trigonometric functions. Main article: Differentiation under the integral sign. Main article: General Leibniz rule.
Ayres, E. Wrede, M. Speigel, S. Lipschutz, J. Schiller, D. Math Vault. Retrieved Antiderivative Arc length Basic properties Constant of integration Fundamental theorem of calculus Differentiating under the integral sign Integration by parts Integration by substitution trigonometric Euler Weierstrass Partial fractions in integration Quadratic integral Trapezoidal rule Volumes Washer method Shell method. Divergence theorem Geometric Hessian matrix Jacobian matrix and determinant Lagrange multiplier Line integral Matrix Multiple integral Partial derivative Surface integral Volume integral Advanced topics Differential forms Exterior derivative Generalized Stokes' theorem Tensor calculus.
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Antiderivative Integral improper Riemann integral Lebesgue integration Contour integration Integral of inverse functions. Parts Discs Cylindrical shells Substitution trigonometric , Weierstrass , Euler Euler's formula Partial fractions Changing order Reduction formulae Differentiating under the integral sign Risch algorithm.
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