Basic Integration... How? (NancyPi)

Introduction

This tutorial is designed to guide you through the fundamentals of basic integration, focusing on how to find antiderivatives or indefinite integrals using essential integration rules. Understanding these concepts is crucial for tackling calculus problems and applying them in various mathematical contexts.

Step 1: Using the Power Rule for Polynomials

The Power Rule is a primary method for integrating polynomial expressions. Follow these steps:

1. Identify the term: Look for terms like x^n where n is a real number.
2. Apply the Power Rule:
   • Increase the power by 1.
   • Divide by the new power.
   • For example, the integral of x^3 is computed as:

     ∫x^3 dx = (x^(3+1))/(3+1) = x^4/4
     

3. Handle constants: If there’s a constant multiplied by the term, keep the constant and integrate the power separately. For example:

   ∫6x^2 dx = 6 * (x^(2+1))/(2+1) = 6 * (x^3/3) = 2x^3
   

4. Sum and Difference: Combine results from multiple terms using addition or subtraction.
5. Add the constant of integration: Always include "+ c" at the end of your integral result since it’s an indefinite integral.

Step 2: Integrating Negative and Fractional Powers

You can still use the Power Rule for negative and fractional powers, with adjustments:

1. Negative Powers: Integrate using the Power Rule, ensuring the power is not -1.
2. Fractional Powers: Increase the power by 1, and simplify if necessary.
   • For example, the integral of x^(1/2) is:

     ∫x^(1/2) dx = (x^(1/2 + 1))/(1/2 + 1) = x^(3/2)/(3/2) = (2/3)x^(3/2)
     

3. Radicals: Rewrite roots as fractional powers before applying the Power Rule.

Step 3: Applying the Log Rule for x^(-1)

When integrating the function x^(-1) or 1/x, use the Log Rule:

1. Identify the function: Look for x^(-1) or 1/x.
2. Apply the Log Rule:

   ∫(1/x) dx = ln|x| + c
   

Step 4: Simplifying Before Integration

Sometimes you’ll need to manipulate the integrand for easier integration:

1. Separate fractions: If integrating a rational expression, break it into simpler fractions.
2. Expand products: Multiply out expressions to convert them into polynomial form.
   • For example, integrate x(x + 1):

     ∫x(x + 1) dx = ∫(x^2 + x) dx
     

Step 5: Using Trigonometric and Exponential Rules

For trigonometric and exponential functions, reference the Table of Integrals:

1. Look up integration rules: Identify the integral form in the table.
2. Use trigonometric identities: If a function isn’t directly integrable, apply identities to rewrite it in a more manageable form.

Conclusion

In summary, mastering basic integration involves understanding and applying the Power Rule, Log Rule, and additional algebraic techniques to manipulate expressions before integrating. To continue your learning journey, consider exploring more advanced integration techniques like U-Substitution or Integration by Parts. Happy integrating!