Calculate Integral with Steps

What is an Integral?

An integral represents the area under a curve between two points on the x-axis. It can be calculated as the limit of a Riemann sum, where the area is approximated by rectangles and the number of rectangles approaches infinity.

There are two main types of integrals:

• Definite Integral: Calculates the exact area under a curve between specific limits (a and b).
• Indefinite Integral: Finds the antiderivative of a function, representing a family of curves.

How to Calculate an Integral

Calculating integrals involves applying integration rules and techniques. Here’s a step-by-step approach:

1. Identify the type of integral: Determine if it's definite or indefinite.
2. Apply basic rules: Use power rule, constant multiple rule, and sum/difference rule.
3. Use substitution for complex functions: Substitute u = g(x) to simplify the integral.
4. Integrate by parts for products of functions: Use integration by parts formula.
5. Evaluate definite integrals: Apply limits of integration to the antiderivative.

Methods for Calculating Integrals

Basic Integration Rules

These rules form the foundation of integral calculation:

• Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1.
• Constant Multiple Rule: ∫k*f(x) dx = k*∫f(x) dx.
• Sum/Difference Rule: ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.

Substitution Method

The substitution method (u-substitution) simplifies integrals by changing variables. Steps:

1. Choose u = g(x).
2. Find du/dx and solve for dx.
3. Rewrite the integral in terms of u.
4. Integrate with respect to u.
5. Substitute back in terms of x.

Worked Examples

Example 1: Basic Power Rule

Calculate ∫x^2 dx.

Solution:

1. Apply the power rule: ∫x^2 dx = (x^(2+1))/(2+1) + C = x^3/3 + C.
2. Final answer: x^3/3 + C.

Example 2: Substitution Method

Calculate ∫2x*e^(x^2) dx.

Solution:

1. Let u = x^2, then du/dx = 2x ⇒ du = 2x dx.
2. Rewrite the integral: ∫e^u du.
3. Integrate: e^u + C.
4. Substitute back: e^(x^2) + C.