1. What Is Integral Calculus?

Integral calculus is a branch of mathematics focused on finding the antiderivatives of functions and calculating areas under curves. It's the inverse operation of differentiation.

2. Power Rule Explanation

The power rule for integration states:

Where:

• \( x \) — Variable of integration
• \( n \) — Exponent (any real number except -1)
• \( C \) — Constant of integration

Special Case: When \( n = -1 \), the integral becomes \( \ln|x| + C \).

3. Practical Applications

Applications: Calculating areas, volumes, work done in physics, probability distributions, and solving differential equations.

4. Using the Calculator

Instructions: Enter the exponent value (n) to calculate the indefinite integral of x^n. The calculator handles all real numbers except n = -1 (which returns natural log).

5. Frequently Asked Questions (FAQ)

Q1: Why is n = -1 a special case?
A: Because the denominator would become zero, and the antiderivative follows a logarithmic pattern instead.

Q2: What does the +C mean?
A: It represents the constant of integration, acknowledging there are infinitely many antiderivatives differing by a constant.

Q3: Can this calculate definite integrals?
A: This calculator shows the indefinite integral. For definite integrals, you would evaluate at bounds after finding the antiderivative.

Q4: What about more complex functions?
A: This implements only the basic power rule. Other techniques (substitution, parts) are needed for more complex integrands.

Q5: How accurate is this for fractional exponents?
A: The power rule works for all real exponents (except -1), including fractions and irrational numbers.