# Integrals

## Accumulations of change introduction

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Introduction to integral calculusDefinite integrals introExploring accumulation of changeWorked example: accumulation of change
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## Approximation with Riemann sums

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Riemann approximation introductionOver- and under-estimation of Riemann sumsLeft & right Riemann sumsWorked example: finding a Riemann sum using a tableWorked example: over- and under-estimation of Riemann sumsMidpoint sumsTrapezoidal sumsUnderstanding the trapezoidal ruleRiemann sums reviewMotion problem with Riemann sum approximation
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Left & right Riemann sumsGet 3 of 4 questions to level up!
Over- and under-estimation of Riemann sumsGet 3 of 4 questions to level up!
Midpoint & trapezoidal sumsGet 3 of 4 questions to level up!

## Summation notation review

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Summation notationSummation notationWorked examples: Summation notation
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Summation notationGet 3 of 4 questions to level up!

## Riemann sums in summation notation

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Riemann sums in summation notationRiemann sums in summation notationWorked example: Riemann sums in summation notationMidpoint and trapezoidal sums in summation notationRiemann sums in summation notation: challenge problem
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Riemann sums in summation notationGet 3 of 4 questions to level up!

## Defining integrals with Riemann sums

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Definite integral as the limit of a Riemann sumDefinite integral as the limit of a Riemann sumWorked example: Rewriting definite integral as limit of Riemann sumWorked example: Rewriting limit of Riemann sum as definite integral
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Definite integral as the limit of a Riemann sumGet 3 of 4 questions to level up!

## Fundamental theorem of calculus and accumulation functions

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The fundamental theorem of calculus and accumulation functionsFunctions defined by definite integrals (accumulation functions)Finding derivative with fundamental theorem of calculusFinding derivative with fundamental theorem of calculus: chain rule

## Interpreting the behavior of accumulation functions

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Interpreting the behavior of accumulation functionsInterpreting the behavior of accumulation functions
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## Properties of definite integrals

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Negative definite integralsFinding definite integrals using area formulasDefinite integral over a single pointIntegrating scaled version of functionSwitching bounds of definite integralIntegrating sums of functionsWorked examples: Finding definite integrals using algebraic propertiesDefinite integrals on adjacent intervalsWorked example: Breaking up the integral's intervalWorked example: Merging definite integrals over adjacent intervalsFunctions defined by integrals: switched intervalFinding derivative with fundamental theorem of calculus: x is on lower boundFinding derivative with fundamental theorem of calculus: x is on both boundsFunctions defined by integrals: challenge problemDefinite integrals properties review
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Finding definite integrals using area formulasGet 3 of 4 questions to level up!
Definite integrals over adjacent intervalsGet 3 of 4 questions to level up!

## Fundamental theorem of calculus and definite integrals

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The fundamental theorem of calculus and definite integralsAntiderivatives and indefinite integralsProof of fundamental theorem of calculus
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Antiderivatives and indefinite integralsGet 3 of 4 questions to level up!

## Reverse power rule

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Reverse power ruleIndefinite integrals: sums & multiplesRewriting before integratingRewriting before integrating: challenge problemReverse power rule review
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Reverse power ruleGet 3 of 4 questions to level up!
Reverse power rule: sums & multiplesGet 3 of 4 questions to level up!
Reverse power rule: rewriting before integratingGet 3 of 4 questions to level up!

## Indefinite integrals of common functions

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Indefinite integral of 1/xIndefinite integrals of sin(x), cos(x), and eˣCommon integrals review
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Indefinite integrals: eˣ & 1/xGet 3 of 4 questions to level up!
Indefinite integrals: sin & cosGet 3 of 4 questions to level up!
Integrating trig functionsGet 5 of 7 questions to level up!

## Definite integrals of common functions

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Definite integrals: reverse power ruleDefinite integral of rational functionDefinite integral of radical functionDefinite integral of trig functionDefinite integral involving natural logDefinite integral of piecewise functionDefinite integral of absolute value function
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Definite integrals: reverse power ruleGet 3 of 4 questions to level up!
Definite integrals: common functionsGet 3 of 4 questions to level up!
Definite integrals of piecewise functionsGet 3 of 4 questions to level up!

## Integrating with u-substitution

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𝘶-substitution intro𝘶-substitution: multiplying by a constant𝘶-substitution: defining 𝘶𝘶-substitution: defining 𝘶 (more examples)𝘶-substitution𝘶-substitution: rational function𝘶-substitution: logarithmic function𝘶-substitution warmup𝘶-substitution: definite integrals𝘶-substitution with definite integrals𝘶-substitution: definite integral of exponential function𝘶-substitution: special application𝘶-substitution: double substitution𝘶-substitution: challenging application
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𝘶-substitution: defining 𝘶Get 3 of 4 questions to level up!
𝘶-substitution: indefinite integralsGet 3 of 4 questions to level up!
𝘶-substitution: definite integralsGet 3 of 4 questions to level up!

## Integrating using long division and completing the square

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Integration using long divisionIntegration using completing the square and the derivative of arctan(x)
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Integration using long divisionGet 3 of 4 questions to level up!
Integration using completing the squareGet 3 of 4 questions to level up!

## Integrating using trigonometric identities

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Integral of cos^3(x)Integral of sin^2(x) cos^3(x)Integral of sin^4(x)
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Integration using trigonometric identitiesGet 3 of 4 questions to level up!

## Trigonometric substitution

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Introduction to trigonometric substitutionSubstitution with x=sin(theta)More trig sub practiceTrig and u substitution together (part 1)Trig and u substitution together (part 2)Trig substitution with tangentMore trig substitution with tangentLong trig sub problem
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Trigonometric substitutionGet 3 of 4 questions to level up!

## Integration by parts

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Integration by parts introIntegration by parts: ∫x⋅cos(x)dxIntegration by parts: ∫ln(x)dxIntegration by parts: ∫x²⋅𝑒ˣdxIntegration by parts: ∫𝑒ˣ⋅cos(x)dxIntegration by parts: definite integralsIntegration by parts challengeIntegration by parts review
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Integration by partsGet 3 of 4 questions to level up!
Integration by parts: definite integralsGet 3 of 4 questions to level up!

## Integrating using linear partial fractions

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Integration with partial fractions
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Integration with partial fractionsGet 3 of 4 questions to level up!

## Improper integrals

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Introduction to improper integralsDivergent improper integralImproper integrals reviewImproper integral with two infinite bounds
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Improper integralsGet 3 of 4 questions to level up!

## Proof videos

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Proof of fundamental theorem of calculusIntuition for second part of fundamental theorem of calculus