An integration formula sheet is useful only when you understand when each rule applies. Memorizing a table without recognizing structure can lead to wrong methods. This guide summarizes common antiderivative rules and shows how to use them as a decision tool rather than a shortcut list.
Power and constant rules
For \(n\ne -1\), \(\int x^n\,dx=x^{n+1}/(n+1)+C\). Constants factor out: \(\int cf(x)\,dx=c\int f(x)\,dx\). Sums split term by term.
Worked example 1: polynomial
Evaluate \(\int (4x^3-6x+2)\,dx\). Use the power rule term by term: \(x^4-3x^2+2x+C\). The constant term becomes \(2x\), and the final answer includes \(+C\).
Worked example 2: logarithmic exception
Evaluate \(\int 5/x\,dx\). The power rule exception applies because \(x^{-1}\) would require division by zero in the power formula. The result is \(5\ln|x|+C\).
Worked example 3: trigonometric rule
Evaluate \(\int (3\cos x-2\sin x)\,dx\). The antiderivative is \(3\sin x+2\cos x+C\). The sign on the sine integral matters because \(\int \sin x\,dx=-\cos x\).
Method formulas
Substitution uses \(u=g(x)\) and \(du=g'(x)\,dx\). Integration by parts uses \(\int u\,dv=uv-\int v\,du\). Partial fractions decompose rational functions before integration. Trig substitution uses identities to simplify radicals.
Common mistakes
Students often use the power rule on \(1/x\), forget absolute value in logarithmic forms, or apply term-by-term splitting to products. A formula sheet should never replace reading the expression structure.
Calculator support
Use the Integral Calculator to check answers. For specific methods, review U-Substitution, Integration by Parts, and Partial Fractions.
Manual practice set
Turn the formula sheet into a quiz. Cover the right column and identify the rule from the integrand. Use powers, reciprocal functions, exponentials, sines, cosines, and tangent-secant forms. Then reverse the exercise: start with an antiderivative and differentiate it to recover the integrand.
Create mixed problems such as \(\int (3x^2+2e^x-\sin x)\,dx\). A formula sheet is most useful when you can combine multiple simple rules in one problem. Write each term separately before simplifying the final answer.
Choosing formulas responsibly
Do not apply formulas by visual similarity alone. The formula for \(\int 1/x\,dx\) is different from the power rule. The formula for \(\int e^{kx}\,dx\) requires a constant adjustment. Trigonometric integrals often depend on signs.
Human review checklist
Check the variable, constants, signs, absolute values, and \(+C\). If the expression is a product, quotient, or composition, a basic table rule may not apply directly. In that case, choose a method such as substitution, parts, or partial fractions.
Internal study path
Use this sheet as the hub, then move to method pages: U-Substitution, Integration by Parts, Trigonometric Substitution, and Partial Fractions.
Classroom-style activity
Ask students to annotate each formula with a condition. The power rule needs \(n\ne -1\). The logarithmic rule needs absolute value unless the inside is known positive. Exponential rules with constants need constant adjustment. Writing conditions beside formulas prevents blind substitution.
Then create mixed exercises that require one direct formula and one method formula. For example, \(\int (2x+e^{3x})\,dx\) uses a direct power rule and a constant-adjusted exponential rule. More complex examples can then move into substitution or parts.
For human review, confirm that every formula matches the expression structure. Formula sheets are dangerous when products, quotients, or compositions are mistaken for simple terms. A good cheat sheet should point users toward the correct method when a direct rule does not apply.
Editorial quality gate
This guide is designed to be used with a human check, not as a blind answer source. Before relying on any result, review the notation, the variable, the assumptions, and the final answer type. If a formula contains a bound, a domain restriction, an absolute value, or a convergence condition, that detail should appear in the written reasoning, not only in the final line.
For independent verification, use at least one manual test. Differentiate antiderivatives, substitute endpoints in definite integrals, compare signs with a quick graph estimate, or test a limit by direct substitution before using a special rule. When a calculator result disagrees with handwritten work, first check whether the input was interpreted correctly. Parentheses, variables, and bounds can change the entire problem.
The related calculator link is included for practice and comparison. A productive study workflow is to try the setup on paper, read the guide section that matches the method, run the calculator, and then explain any difference in your own words. That final explanation step is what turns a solved example into durable calculus understanding.
Before you move on
Close the guide by writing one original problem, one method clue, and one verification step. For example, the clue might be an inside derivative, an indeterminate form, a repeated factor, a variable bound, or a physical unit. The verification step might be differentiation, substitution, estimation, graph behavior, or comparison with a known formula. This short habit helps the article become active practice rather than passive reading.
For best results, revisit the examples after a break and solve them without looking at the steps. If the method still feels clear, the guide has done its job.