Calculus: reverse the chain rule
Integration by Substitution
Identify the inner function and its derivative before substituting. Differentiate the result to check.
Download Integration by Substitution PDF + answersCollege math practice
College math worksheets
Practice calculus, statistics, matrices, vectors, and linear algebra. Check the sample notation and problem type against what your course covers. Work through the setup by hand before simplifying or calculating.
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Each PDF includes a worksheet and matching answer key. Choose the skill and level that fit the learner; these are individual practice sets, not a complete course.
Download Integration by Substitution PDF + answers
Multivariable calculus
Partial Derivatives
Differentiate with respect to one variable while treating the others as constants.
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Statistical inference
Confidence Intervals for Means
Practice interval calculations and interpret the interval in the context of a population mean.
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College math topics
30 level-focused worksheet skills lead this collection, with 80 prerequisite and review skills listed separately. Choose a topic to see its suggested setup in the builder.
Understand data and chance
8 skills- Binomial ProbabilitiesCalculate exact probabilities for binomial random variables.
- Sampling DistributionsFind the mean and standard error of a sample-mean distribution.
- Confidence Intervals for ProportionsConstruct exact simplified one-proportion z intervals in structured settings.
- Confidence Intervals for MeansConstruct exact z intervals for a mean when the population standard deviation is known.
- One-Sample Hypothesis TestsCalculate a one-sample z statistic and make the stated two-sided decision.
- Two-Sample Mean InferenceCompute an exact two-sample z statistic from independent sample means and known standard errors.
- Chi-Square Goodness of FitCompute exact chi-square statistics for two-category goodness-of-fit tables.
- One-Way ANOVACompute exact one-way ANOVA F statistics for three balanced structured groups.
Study change and accumulation
12 skills- Integration by SubstitutionEvaluate indefinite integrals that reverse the chain rule with an exact substitution.
- Integration by PartsEvaluate exact definite integrals of polynomial–exponential products by parts.
- Partial-Fraction DecompositionDecompose rational functions with two distinct linear factors into exact partial fractions.
- Improper IntegralsDecide convergence and evaluate exact power-tail improper integrals when they converge.
- Infinite Geometric SeriesRecognize convergent geometric series and calculate their exact sums.
- Maclaurin-Series CoefficientsFind exact coefficients in the Maclaurin series of exponential, sine, and cosine functions.
- Parametric DerivativesFind exact tangent slopes for polynomial parametric curves.
- Polar and Cartesian CoordinatesConvert exact axis-aligned points between polar and Cartesian coordinates.
- Partial DerivativesEvaluate exact first partial derivatives of bivariate polynomials.
- Gradients and Directional DerivativesUse an exact gradient and a unit direction vector to find a directional derivative.
- Tangent PlanesWrite exact tangent-plane equations for quadratic surfaces.
- Double Integrals over RectanglesEvaluate exact iterated integrals of linear functions over rectangular regions.
Work with matrices and vectors
10 skills- Row ReductionRow-reduce augmented matrices for two-variable systems with unique integer solutions.
- Eigenvalues and EigenvectorsFind exact eigenvalues and a valid eigenvector for each eigenvalue of structured 2 by 2 matrices.
- DeterminantsEvaluate exact determinants of two-by-two matrices and connect them to invertibility.
- Two-by-Two Matrix InversesFind exact inverses of structured two-by-two matrices with determinant one.
- Dot Products and OrthogonalityCompute dot products and use a zero dot product to recognize orthogonal vectors.
- Vector ProjectionsProject a two-dimensional vector exactly onto a diagonal direction vector.
- Gram–Schmidt OrthogonalizationCarry out an exact Gram–Schmidt step for a structured pair of independent vectors.
- Independence, Basis, and DimensionDetermine whether two planar vectors form a basis and state the dimension of their span.
- Change of Basis CoordinatesFind coordinate vectors relative to a nonstandard ordered basis in R².
- Diagonalization and Matrix PowersUse the eigenvalues of a symmetric two-by-two matrix to compute an exact matrix power.
Prerequisite and review skillsUseful practice whose primary level is another grade range.80 skills
Solve equations and use functions
35 skills- Multi-Step EquationsSolve linear equations requiring distribution and more than one inverse operation.
- Variables on Both SidesSolve linear equations with variable terms on both sides.
- Literal EquationsRearrange formulas to isolate a requested variable.
- Absolute-Value EquationsSolve absolute-value equations and identify their two solutions.
- Function NotationEvaluate functions and interpret function notation.
- Domain & RangeFind domain and range from relations, tables, mappings, formulas, and graphs.
- Systems by SubstitutionSolve two-variable systems using substitution.
- Systems by EliminationSolve two-variable systems using elimination.
- Systems of InequalitiesTest whether ordered pairs satisfy both inequalities in a system.
- Interpreting Parabola FeaturesIdentify a parabola's minimum value and where it occurs from its vertex.
- Average Rate of ChangeCalculate average rate of change over an interval.
- Comparing FunctionsCompare rates of change across point and equation representations.
- Function TransformationsDescribe vertical stretches, reflections, and translations of the quadratic parent function.
- Piecewise FunctionsEvaluate piecewise-defined functions at boundary and non-boundary inputs.
- Function CompositionEvaluate compositions of two functions at given input values.
- Inverse FunctionsFind inverse rules for one-to-one linear functions.
- Linear vs. Exponential ModelsDistinguish constant differences from constant ratios.
- Exponential Growth and DecayCalculate repeated percent growth or decay over a given number of periods.
- Compound InterestCalculate compound interest using standard formulas.
- Logarithm PropertiesExpand logarithmic expressions using the power and quotient properties.
- Polynomial OperationsAdd, subtract, multiply, divide, and evaluate polynomials.
- Polynomial Long DivisionDivide polynomials by exact linear factors and write each quotient.
- Synthetic DivisionUse synthetic division with a linear divisor.
- Remainder and Factor TheoremsConnect polynomial evaluation, remainders, and factors.
- Polynomial Zeros and MultiplicityRead zeros and multiplicities from factored polynomials.
- Radical ExpressionsSimplify, operate on, rationalize, and solve radical expressions.
- Radical EquationsSolve radical equations and check for extraneous solutions.
- Rational ExpressionsSimplify, operate on, and solve rational expressions.
- Rational EquationsSolve equations containing rational expressions and state excluded values.
- Quadratic FactoringFactor monic quadratic expressions with integer roots.
- Completing the SquareRewrite quadratics in completed-square form.
- Quadratic FormulaSolve quadratic equations using the quadratic formula.
- Discriminant ClassificationUse the discriminant to classify quadratic roots.
- Quadratic Solutions & GraphsSolve quadratics and connect roots, vertices, discriminants, and graphs.
- Linear-Quadratic SystemsFind intersections of a line and a quadratic curve.
Understand data and chance
22 skills- Mean, Median, Mode & RangeFind the mean, median, mode, or range for small data sets.
- Charts & Graph ComprehensionRead bar graphs, line graphs, and circle graphs.
- Basic ProbabilityFind simple probabilities from counts.
- Standard DeviationCalculate and interpret population standard deviation.
- Distribution Shape and OutliersDescribe distribution shape and effects of outliers.
- Normal Distributions and Z-ScoresCompute z-scores and use normal-distribution percentages.
- Two-Way Frequency TablesCalculate joint, marginal, and conditional relative frequencies.
- ScatterplotsDescribe direction, form, strength, and unusual features in scatterplots.
- Correlation CoefficientInterpret the sign and strength of correlation coefficients.
- Regression LinesUse and interpret least-squares regression equations.
- ResidualsCalculate and interpret residuals from model predictions.
- Correlation vs. CausationDistinguish association from evidence of causation.
- Compound EventsRepresent unions, intersections, and complements of events.
- Conditional ProbabilityCalculate and interpret conditional probability.
- Independent EventsDetermine whether events are independent using probabilities.
- Probability RulesApply addition and multiplication rules of probability.
- Permutations and CombinationsCount ordered and unordered selections.
- Expected ValueCompute expected values and compare decisions.
- Sampling Methods and BiasRecognize representative samples and common sources of bias.
- Types of Statistical StudiesDistinguish surveys, observational studies, and experiments.
- Margin of ErrorInterpret confidence intervals and margins of error.
- Simulation-Based InferenceUse simulated results to judge whether an outcome is surprising.
Use exponents and complex numbers
1 skillStudy change and accumulation
8 skills- Advanced Derivative RulesDifferentiate with product, quotient, chain, exponential, logarithmic, and trig rules.
- Definite IntegrationEvaluate definite integrals and basic area applications.
- Intro Differential EquationsSolve basic rate, growth, decay, and separable differential equations.
- Power Rule DerivativesDifferentiate polynomial terms with the power rule.
- Basic Power IntegralsIntegrate polynomial terms with the power rule.
- Polynomial LimitsEvaluate limits of polynomial expressions by substitution.
- Applications of DerivativesApply derivatives to velocity, tangent slopes, extrema, and optimization.
- Applications of IntegrationApply definite integrals to net change, area, average value, and volumes of revolution.
Work with matrices and vectors
4 skillsUse trigonometry and periodic functions
10 skills- Right-Triangle Trigonometric RatiosFind sine, cosine, and tangent ratios in right triangles.
- Missing Sides and AnglesUse trigonometric ratios to solve right triangles.
- Degrees and RadiansConvert angle measures between degrees and radians.
- Unit-Circle ValuesRecall exact sine and cosine values for special angles.
- Reference AnglesFind reference angles and signs by quadrant.
- Graphs of Sine and CosineIdentify key points, amplitude, midline, and period of basic trig graphs.
- Trig Function TransformationsAnalyze amplitude, frequency, phase shift, and midline.
- Inverse Trigonometric FunctionsUse inverse trigonometric functions to find angles.
- Trigonometric IdentitiesApply fundamental trigonometric identities.
- Law of Sines and CosinesSolve non-right triangles using the Laws of Sines and Cosines.
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