JEE Main & Advanced Mathematics Indefinite Integrals Definition

Definition

Category : JEE Main & Advanced

A function \[\varphi (x)\] is called a primitive or an antiderivative of a function \[f(x)\], if \[\varphi '(x)=f(x).\]

Let \[f\,(x)\] be a function. Then the collection of all its primitives is called the indefinite integral of \[f(x)\] and is denoted by \[\int{f(x)\,dx.}\]

Thus, \[\frac{d}{dx}(\varphi (x)+c)=f(x)\Rightarrow \int{f(x)\,dx}=\varphi (x)+c\]

where \[\varphi (x)\] is primitive of \[f(x)\] and c is an arbitrary constant known as the constant of integration.

Other Topics

Definition
Fundamental Integration Formulae
Integration By Substitution
Integration by Parts
Evaluation of the Various Forms of Integrals by Use of Standard Results
Integrals of the Form \[\int{\frac{dx}{a+b\,cos\,x}}\] and \[\int{\frac{dx}{a+b\,sin\,x}}\]
Integrals of the Form \[\int{\frac{dx}{a+bcosx+csinx}}\,,\,\,\int{\frac{dx}{asinx+bcosx}}\]
Integrals of the form \[\int_{{}}^{{}}{\frac{dx}{a+b\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x},\,\int_{{}}^{{}}{\frac{dx}{a+b\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x},}}\]\[\int_{{}}^{{}}{\frac{dx}{a\,\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+b\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}\mathbf{,}}\]\[\int_{{}}^{{}}{\frac{dx}{{{\mathbf{(}a\,\sin x+b\,\mathbf{cos}\,x\mathbf{)}}^{\mathbf{2}}}}\mathbf{,}}\int_{{}}^{{}}{\frac{dx}{a\,+b\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}x+c\,\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}x}}\]
Integrals of the form \[\int{\frac{a sinx+b cosx}{c sinx+d cosx} }\]and \[\int{\frac{a\,sinx+bcos\,x+q}{c\,sinx+d\,cosx+r}}\]
Integration of Rational Functions by Using Partial Fractions
Integration of Trigonometric Functions
Integration of Hyperbolic Functions