Integral Calculus Cheatsheet

Cheat Codes & Shortcuts

• Definition: Integral represents area under a curve or antiderivative.
• Indefinite Integral: \( \int f(x) \, dx = F(x) + C \), where \( F'(x) = f(x) \).
• Definite Integral: \( \int_a^b f(x) \, dx = F(b) - F(a) \).
• Substitution Rule: \( \int f(g(x)) g'(x) \, dx = \int f(u) \, du \), where \( u = g(x) \).
• Integration by Parts: \( \int u \, dv = uv - \int v \, du \).
• Partial Fractions: Decompose rational functions for integration.
• Trigonometric Integrals: Use identities like \( \sin^2 x = \frac{1 - \cos 2x}{2} \).
• Improper Integrals: Evaluate limits for infinite bounds or discontinuities.
• Area Between Curves: \( \int_a^b [f(x) - g(x)] \, dx \).
• Fundamental Theorem: If \( F'(x) = f(x) \), then \( \int_a^b f(x) \, dx = F(b) - F(a) \).

Quick Reference Table

Type	Form	Solution
Basic	\( \int x^n \, dx \)	\( \frac{x^{n+1}}{n+1} + C \), \( n \neq -1 \)
Substitution	\( \int x e^{x^2} \, dx \)	Let \( u = x^2 \), then \( \int \frac{1}{2} e^u \, du \)
By Parts	\( \int x e^x \, dx \)	Use \( u = x \), \( dv = e^x \, dx \)
Trigonometric	\( \int \sin^2 x \, dx \)	Use \( \sin^2 x = \frac{1 - \cos 2x}{2} \)
Partial Fractions	\( \int \frac{1}{x^2 - 1} \, dx \)	Decompose as \( \frac{A}{x-1} + \frac{B}{x+1} \)
Improper	\( \int_1^\infty \frac{1}{x^2} \, dx \)	Evaluate \( \lim_{t \to \infty} \int_1^t \frac{1}{x^2} \, dx \)

Advice

Integral Calculus Quick Tips

• Basic Integrals: Memorize forms like \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
• Substitution: Choose \( u \) to eliminate complex expressions.
• Integration by Parts: Use when product of functions, prioritize \( u \) that simplifies.
• Trigonometric Integrals: Use identities like \( \cos^2 x = \frac{1 + \cos 2x}{2} \).
• Improper Integrals: Check convergence by evaluating limits.

Integral Calculus Speed Quiz

Test your speed with 5 integral calculus questions! You have 30 seconds per question.