Properties of Indefinite Integrals
1. Linearity of Indefinite Integrals:
 Property: The integral of a sum/difference is the sum/difference of the integrals.
 $\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$
 $\int [f(x)g(x)]dx=\int f(x)dx\int g(x)dx$
 This property holds true for constants as well: $\int kf(x)dx=k\int f(x)dx$.
2. Integral of a Constant:
 Property: The integral of a constant is the constant multiplied by the variable.
 $\int kdx=kx+C$ (where $k$ is a constant and $C$ is the constant of integration).
3. Integration of Zero:
 Property: The integral of zero with respect to any variable is a constant.
 $\int 0\text{\hspace{0.17em}}dx=C$ (where $C$ is the constant of integration).
4. Reciprocal Rule:
 Property: The integral of the reciprocal function $1\mathrm{/}f(x)$ is the logarithm of the absolute value of $f(x)$.
 $\int \frac{1}{f(x)}dx=\mathrm{ln}\mathrm{\mid}f(x)\mathrm{\mid}+C$.
5. Integration by Substitution:
 Property: Allows for the substitution of variables to simplify integrals.
 The integral of $f(u)\cdot \frac{du}{dx}$ with respect to $u$ equals the integral of $f(u)$ with respect to $u$.
 This allows solving integrals by substituting $u$ for a more manageable expression.
6. Integration of Composite Functions:
 Property: Involves integrating composite functions.
 The integral of a composite function $f(g(x))$ may require techniques like substitution or integration by parts.
7. Addition of Arbitrary Constant (Constant of Integration):
 Property: Indefinite integrals have an arbitrary constant of integration denoted by $C$.
 $\int f(x)dx=F(x)+C$, where $F(x)$ is the antiderivative or integral function.
8. Change of Variables:
 Statement: If u = g(x) is a differentiable function, then ∫ f(g(x)) * g'(x) dx = ∫ f(u) du.
 This property is fundamental in substitution method, where a change of variable simplifies the integral.
9. Additive Property of Limits:
 Statement: If F(x) is an antiderivative of f(x) on an interval [a, b], then ∫ f(x) dx from a to b = F(b)  F(a).
 This property relates the definite integral to the antiderivative of the integrand.
10. Integral of a Derivative:
 Statement: The integral of the derivative of a function is the original function, up to a constant.
 ∫ f'(x) dx = f(x) + C
11. Integration by Parts:
 Statement: Integration by parts is based on the formula: ∫ u dv = uv  ∫ v du.
 Useful for integrating the product of two functions.
Importance of Properties:

Simplification: Properties help simplify complex integrals, breaking them down into manageable forms.

Problemsolving: Understanding these properties aids in solving a wide range of integral problems efficiently.

Fundamental Tools: These properties form the foundation for further exploration in calculus and its applications.