# Properties of Indefinite Integrals

### 1. Linearity of Indefinite Integrals:

• Property: The integral of a sum/difference is the sum/difference of the integrals.
• $\int \left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx$
• $\int \left[f\left(x\right)-g\left(x\right)\right]dx=\int f\left(x\right)dx-\int g\left(x\right)dx$
• This property holds true for constants as well: $\int kf\left(x\right)dx=k\int f\left(x\right)dx$.

### 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 $1\mathrm{/}f\left(x\right)$ is the logarithm of the absolute value of $f\left(x\right)$.
• $\int \frac{1}{f\left(x\right)}dx=\mathrm{ln}\mathrm{\mid }f\left(x\right)\mathrm{\mid }+C$.

### 5. Integration by Substitution:

• Property: Allows for the substitution of variables to simplify integrals.
• The integral of $f\left(u\right)\cdot \frac{du}{dx}$ with respect to $u$ equals the integral of $f\left(u\right)$ 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 $f\left(g\left(x\right)\right)$ 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 $C$.
• $\int f\left(x\right)dx=F\left(x\right)+C$, where $F\left(x\right)$ 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.

• Problem-solving: 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.