Indefinite Intergral

Indefinite integrals

Introduction to indefinite integrals

The indefinite integral of a function is the collection of all its antiderivatives. It represents a family of functions whose derivatives equal the integrand. The general form is:

$\int f(x)\,dx = F(x) + C$

Where:

∫ is the integral symbol.

f(x) is the function to be integrated.

dx indicates the variable of integration.

F(x) is the antiderivative of f(x).

C is the constant of integration.

Basic rules of indefinite integration

1. Power rule:

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad (n \neq -1)$

2. Constant multiple rule:

$\int a \cdot f(x)\,dx = a \int f(x)\,dx$

3. Sum and difference rule:

$\int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx$

4. Exponential rule:

$\int e^x\,dx = e^x + C$

5. Logarithmic rule:

$\int \frac{1}{x}\,dx = \ln|x| + C$

6. Trigonometric functions:

$\int \sin(x)\,dx = -\cos(x) + C$

$\int \cos(x)\,dx = \sin(x) + C$

$\int \sec^2(x)\,dx = \tan(x) + C$

$\int \csc^2(x)\,dx = -\cot(x) + C$

$\int \sec(x)\tan(x)\,dx = \sec(x) + C$

$\int \csc(x)\cot(x)\,dx = -\csc(x) + C$

Step-by-step examples

Example 1: $\int 5x^3\,dx$

1. Apply the constant multiple rule:

   $\int 5x^3\,dx = 5 \int x^3\,dx$

2. Use the power rule:

   $5 \cdot \frac{x^{3+1}}{3+1} + C$

3. Simplify:

   $\frac{5x^4}{4} + C$

4. Final Answer: $\dfrac{5x^4}{4} + C$

Example 2: $\int (3x^2 - 4x + 5)\,dx$

1. Split using the sum/difference rule:

   $\int 3x^2\,dx - \int 4x\,dx + \int 5\,dx$

2. Apply the power rule to each term:

   $\left(3 \cdot \frac{x^3}{3}\right) - \left(4 \cdot \frac{x^2}{2}\right) + (5x) + C$

3. Simplify:

   $x^3 - 2x^2 + 5x + C$

4. Final Answer: $x^3 - 2x^2 + 5x + C$

Example 3: $\int \sin(x)\,dx$

1. Apply the standard rule:

   $\int \sin(x)\,dx = -\cos(x) + C$

2. Final Answer: $-\cos(x) + C$

Example 4: $\int e^{2x}\,dx$

1. Use substitution: Let $u = 2x$, so $du = 2\,dx$ or $dx = du/2$.

2. Rewrite the integral:

   $\int e^u \cdot \frac{du}{2} = \frac{1}{2} \int e^u\,du$

3. Integrate:

   $\frac{1}{2} \cdot e^u + C$

4. Substitute back $u = 2x$:

   $\frac{1}{2} \cdot e^{2x} + C$

5. Final Answer: $\dfrac{1}{2} e^{2x} + C$

Example 5: $\int \dfrac{1}{\sqrt{x}}\,dx$

1. Rewrite the integrand:

   $\int x^{-1/2}\,dx$

2. Apply the power rule:

   $\frac{x^{(-1/2)+1}}{(-1/2)+1} + C$

3. Simplify:

   $\frac{x^{1/2}}{1/2} + C$

4. Invert the denominator:

   $2\sqrt{x} + C$

5. Final Answer: $2\sqrt{x} + C$

Important notes on indefinite integrals

1. The constant of integration C must always be added since integration can result in multiple functions differing by a constant.
2. Indefinite integrals represent families of functions rather than a single solution.
3. More advanced functions may require substitution or other integration techniques.