Fundamentals of Integration

Master anti-differentiation, indefinite integrals, constant of integration + C, and evaluating definite integrals under curves.

Learning Objectives

Understand integration as the reverse process of differentiation
Apply the rule ∫xⁿ dx = (xⁿ⁺¹ / (n + 1)) + C for n ≠ -1
Include the constant of integration C for indefinite integrals
Evaluate definite integrals between limits a and b

Concept Explanation

Integration is the reverse process of differentiation (sometimes called anti-differentiation). While differentiation finds the gradient of a curve, integration enables us to find the original equation of a curve or calculate the area bounded underneath a curve.

Reverse Power Rule: Add 1 to the exponent n, then divide by the new exponent (n + 1).

Constant of Integration (+ C): Because differentiating any constant produces 0, integrating returns a family of parallel curves. We represent the unknown constant with + C.

Definite Integrals: Evaluated between upper limit b and lower limit a. Subtract lower limit value from upper limit value: ∫[a to b] f(x) dx = F(b) - F(a). The constant C cancels out.

Key Mathematical Formulas

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

\int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a)

Worked Example

Step-by-Step Solution

Interactive Walkthrough

Problem Statement:

Evaluate the definite integral:

\int_1^3 (3x^2 + 4x - 2) \, dx

Step 1: Integrate term by term

Add 1 to each power and divide by the new power.

\int (3x^2 + 4x - 2) \, dx = \left[ \frac{3x^3}{3} + \frac{4x^2}{2} - 2x \right]_1^3 = [x^3 + 2x^2 - 2x]_1^3

Final Answer:

38

Try It Yourself

What is the indefinite integral ∫ (6x² + 2x) dx?

\int (6x^2 + 2x) \, dx

Common Student Mistakes

❌ Common Pitfall: Forgetting to write + C on indefinite integrals

✅ Correction: In exam marking schemes, omitting + C loses an automatic accuracy mark for indefinite integrals.

❌ Common Pitfall: Multiplying instead of dividing by the new exponent

✅ Correction: Differentiation multiplies by the power; integration DIVIDES by the new power!

UK Exam Advice & Tips

If a curve lies below the x-axis, its definite integral will be negative. When finding total enclosed area, split the limits where the curve crosses the x-axis and take absolute values.

Lesson FAQ

Why can't we use the power rule for ∫ (1/x) dx?

If you apply the power rule to x⁻¹, you get x⁰ / 0, which is undefined. The integral of 1/x is natural logarithm ln|x| + C.

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