Calculus Review

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Basic Derivative of x to the power of n

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๐‘‘/๐‘‘๐‘ฅ (๐‘ฅ^๐‘›) = ๐‘›๐‘ฅ^(๐‘›โˆ’1)

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Basic Derivative of sin x

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๐‘‘/๐‘‘๐‘ฅ (sin ๐‘ฅ) = cos ๐‘ฅ

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Flashcards of key calculus concepts and theorems.

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102 Terms

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Basic Derivative of x to the power of n

๐‘‘/๐‘‘๐‘ฅ (๐‘ฅ^๐‘›) = ๐‘›๐‘ฅ^(๐‘›โˆ’1)

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Basic Derivative of sin x

๐‘‘/๐‘‘๐‘ฅ (sin ๐‘ฅ) = cos ๐‘ฅ

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Basic Derivative of cos x

๐‘‘/๐‘‘๐‘ฅ (cos ๐‘ฅ) = โˆ’ sin ๐‘ฅ

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Basic Derivative of tan x

๐‘‘/๐‘‘๐‘ฅ (tan ๐‘ฅ) = secยฒ ๐‘ฅ

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Basic Derivative of cot x

๐‘‘/๐‘‘๐‘ฅ (cot ๐‘ฅ) = โˆ’ cscยฒ ๐‘ฅ

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Basic Derivative of sec x

๐‘‘/๐‘‘๐‘ฅ (sec ๐‘ฅ) = sec ๐‘ฅ tan ๐‘ฅ

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Basic Derivative of csc x

๐‘‘/๐‘‘๐‘ฅ (csc ๐‘ฅ) = โˆ’ csc ๐‘ฅ cot ๐‘ฅ

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Basic Derivative of ln u

๐‘‘/๐‘‘๐‘ฅ (ln ๐‘ข) = (1/๐‘ข) * du/dx

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Basic Derivative of e to the power of u

๐‘‘/๐‘‘๐‘ฅ (๐‘’^๐‘ข) = ๐‘’^๐‘ข * du/dx

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Chain Rule

๐‘‘/๐‘‘๐‘ฅ [๐‘“(๐‘ข)] = ๐‘“โ€ฒ(๐‘ข) * du/dx

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Product Rule

๐‘‘/๐‘‘๐‘ฅ (๐‘ข๐‘ฃ) = ๐‘ข * dv/dx + ๐‘ฃ * du/dx

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Quotient Rule

๐‘‘/๐‘‘๐‘ฅ (๐‘ข/๐‘ฃ) = (๐‘ฃ * du/dx - ๐‘ข * dv/dx) / ๐‘ฃยฒ

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Intermediate Value Theorem

If ๐‘“(๐‘ฅ) is continuous on [๐‘Ž, ๐‘], and y is a number between ๐‘“(๐‘Ž) and ๐‘“(๐‘), then there exists at least one number ๐‘ฅ = ๐‘ in (๐‘Ž, ๐‘) such that ๐‘“(๐‘) = ๐‘ฆ

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Mean Value Theorem

If ๐‘“(๐‘ฅ) is continuous on [๐‘Ž, ๐‘], AND the first derivative exists on the interval (๐‘Ž, ๐‘) then there is at least one number ๐‘ฅ = ๐‘ in (๐‘Ž, ๐‘) such that ๐‘“โ€ฒ(๐‘) = (๐‘“(๐‘)โˆ’๐‘“(๐‘Ž))/(๐‘โˆ’๐‘Ž)

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Rolleโ€™s Theorem

If ๐‘“(๐‘ฅ) is continuous on [๐‘Ž, ๐‘], AND the first derivative exists on the interval (๐‘Ž, ๐‘) AND ๐‘“(๐‘Ž) = ๐‘“(๐‘), then there is at least one number ๐‘ฅ = ๐‘ in (๐‘Ž, ๐‘) such that ๐‘“โ€ฒ(๐‘) = 0

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Extreme Value Theorem

If ๐‘“(๐‘ฅ) is continuous on [๐‘Ž, ๐‘], then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.

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Alternate Definition of the Derivative

๐‘“โ€ฒ(๐‘) = lim ๐‘ฅโ†’๐‘ (๐‘“(๐‘ฅ) โˆ’ ๐‘“(๐‘)) / (๐‘ฅ โˆ’ ๐‘)

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Critical Point

Point where dy/dx = 0 or is undefined.

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Local Minimum

๐‘‘๐‘ฆ/๐‘‘๐‘ฅ goes from (โˆ’, 0, +) or (โˆ’, undefined, +) OR ๐‘‘ยฒ๐‘ฆ/๐‘‘๐‘ฅยฒ > 0

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Local Maximum

๐‘‘๐‘ฆ/๐‘‘๐‘ฅ goes (+, 0, โˆ’) or (+, undefined, โˆ’) OR ๐‘‘ยฒ๐‘ฆ/๐‘‘๐‘ฅยฒ < 0

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Point of Inflection

Concavity changes; ๐‘‘ยฒ๐‘ฆ/๐‘‘๐‘ฅยฒ goes from (+, 0, โˆ’), (โˆ’, 0, +), (+, undefined, โˆ’), OR (โˆ’, undefined, +)

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Derivative of an Inverse Function

๐‘”โ€ฒ(๐‘ฅ) = 1 / ๐‘“โ€ฒ(๐‘”(๐‘ฅ))

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Implicit Differentiation

Remember that in implicit differentiation you will have a ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ for each y in the original function or equation. Isolate the ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ. If you are taking the second derivative ๐‘‘ยฒ๐‘ฆ/๐‘‘๐‘ฅยฒ, you will often substitute the expression you found for the first derivative somewhere in the process.

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Average Rate of Change (ARoC)

๐‘š_sec = (๐‘“(๐‘) โˆ’ ๐‘“(๐‘Ž)) / (๐‘ โˆ’ ๐‘Ž)

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Instantaneous Rate of Change (IRoC)

๐‘š_tan = ๐‘“โ€ฒ(๐‘ฅ) = lim โ„Žโ†’0 (๐‘“(๐‘ฅ + โ„Ž) โˆ’ ๐‘“(๐‘ฅ)) / โ„Ž

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Equation of a Tangent Line

y - y1 = m(x - x1)

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๐‘“โ€ฒ(๐‘ฅ) > 0

Function is increasing

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๐‘“โ€ฒ(๐‘ฅ) < 0

Function is decreasing

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๐‘“โ€ฒ(๐‘ฅ) = 0 or DNE

Critical Values at x.

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Relative Maximum

๐‘“โ€ฒ(๐‘ฅ) = 0 or DNE and sign of ๐‘“โ€ฒ(๐‘ฅ) changes from + to โˆ’

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Relative Minimum

๐‘“โ€ฒ(๐‘ฅ) = 0 or DNE and sign of ๐‘“โ€ฒ(๐‘ฅ) changes from โˆ’ to +

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Absolute Max or Min

Check Endpoints Also. The maximum value is a y-value.

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๐‘“โ€ฒโ€ฒ(๐‘ฅ) > 0

Function is concave up.

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๐‘“โ€ฒโ€ฒ(๐‘ฅ) < 0

Function is concave down.

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Point of Inflection Second Derivative

๐‘“โ€ฒ(๐‘ฅ) = 0 and sign of ๐‘“โ€ฒโ€ฒ(๐‘ฅ) changes, then there is a point of inflection at x.

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Relative Maximum (Second derivative)

๐‘“โ€ฒโ€ฒ(๐‘ฅ) < 0

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Relative Minimum (Second derivative)

๐‘“โ€ฒโ€ฒ(๐‘ฅ) > 0

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Horizontal Asymptotes: Largest exponent in numerator < largest exponent in denominator

lim ๐‘ฅโ†’ยฑโˆž ๐‘“(๐‘ฅ) = 0

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Horizontal Asymptotes: Largest exponent in numerator > largest exponent in denominator

lim ๐‘ฅโ†’ยฑโˆž ๐‘“(๐‘ฅ) = DNE

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Horizontal Asymptotes: Largest exponent in numerator = largest exponent in denominator

lim ๐‘ฅโ†’ยฑโˆž ๐‘“(๐‘ฅ) = a/b (quotient of leading coefficients)

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The Fundamental Theorem of Calculus

โˆซ ๐‘“(๐‘ฅ)๐‘‘๐‘ฅ = ๐น(๐‘) โˆ’ ๐น(๐‘Ž) where ๐นโ€ฒ(๐‘ฅ) = ๐‘“(๐‘ฅ)

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Corollary to FTC

๐‘‘/๐‘‘๐‘ฅ โˆซ ๐‘“(๐‘ก)๐‘‘๐‘ก from a to g(x) = ๐‘“(๐‘”(๐‘ฅ)) * ๐‘”โ€ฒ(๐‘ฅ)

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๐‘ฅ(๐‘ก)

Position Function

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๐‘ฃ(๐‘ก)

Velocity Function

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๐‘Ž(๐‘ก)

Acceleration Function

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Relationship between Position, Velocity, and Acceleration (Derivatives)

derivative of position (ft) is velocity (ft/sec); the derivative of velocity (ft/sec) is acceleration (ft/sec2)

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Relationship between Position, Velocity, and Acceleration (Integrals)

The integral of acceleration (ft/sec2) is velocity (ft/sec) ; the integral of velocity (ft/sec) is position (ft).

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Speed

| velocity |

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If acceleration and velocity have the same sign

Then the speed is increasing

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If the acceleration and velocity have different signs

Then the speed is decreasing.

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The particle is moving right

When velocity is positive

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the particle is moving left

When velocity is negative

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Displacement

โˆซ ๐‘ฃ(๐‘ก) ๐‘‘๐‘ก from ๐‘ก0 to ๐‘ก๐‘“

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Total Distance

โˆซ | ๐‘ฃ(๐‘ก)| ๐‘‘๐‘ก from ๐‘ก0 to ๐‘ก๐‘“

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Average Velocity

(final position โˆ’ initial position) / (total time) = ฮ”๐‘ฅ / ฮ”๐‘ก

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Accumulation

๐‘ฅ(0) + โˆซ ๐‘ฃ(๐‘ก) ๐‘‘๐‘ก from ๐‘ก0 to ๐‘ก๐‘“

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Four things on a calculator that need no work shown

Graphing a function within an arbitrary view window, Finding the zeros of a function, Computing the derivative of a function numerically, Computing the definite integral of a function numerically.

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Definition of Logarithms

๐‘™๐‘› ๐‘ = ๐‘ โ†” ๐‘’^๐‘ = ๐‘

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Logarithm Product Rule

๐‘™๐‘› ๐‘€๐‘ = ln ๐‘€ + ln ๐‘

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Logarithm Quotient Rule

ln (๐‘€/๐‘) = ln ๐‘€ โˆ’ ln ๐‘

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Logarithm Power Rule

๐‘ โˆ™ ln ๐‘€ = ln ๐‘€^๐‘

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Exponential Growth and Decay Formula

๐‘ฆ = ๐ถ๐‘’^(๐‘˜๐‘ก)

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โ€œthe rate of change of y is proportional to yโ€

๐‘ฆโ€ฒ = ๐‘˜๐‘ฆ

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When solving a differential equation

Separate variables, Integrate, Add +C to one side, Use initial conditions to find โ€œCโ€, Write the equation if the form of ๐‘ฆ = ๐‘“(๐‘ฅ)

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Mean Value Theorem for Integrals: The Average Value

๐‘“_avg = (1 / (๐‘ โˆ’ ๐‘Ž)) โˆซ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ from a to b

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Riemann Sums

A rectangular approximation. Do NOT EVALUATE THE INTEGRAL; you add up the areas of the rectangles.

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Area of a Trapezoid

๐ด_๐‘‡ = (1/2) โ„Ž[๐‘1 + ๐‘2]

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Trapezoidal Rule (for even intervals)

โˆซ ๐‘“(๐‘ฅ) ๐‘‘๐‘ฅ from a to b = (๐‘ โˆ’ ๐‘Ž) / (2๐‘›) [๐‘ฆ0 + 2๐‘ฆ1 + 2๐‘ฆ2 + โ€ฆ +2๐‘ฆ๐‘›โˆ’1 + ๐‘ฆ๐‘›]

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sin(โˆ’๐‘ฅ) (ODD)

sin(โˆ’๐‘ฅ) = โˆ’ sin ๐‘ฅ

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cos(โˆ’๐‘ฅ) (EVEN)

cos(โˆ’๐‘ฅ) = cos ๐‘ฅ

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Pythagorean Identities

๐‘ ๐‘ ๐‘ ๐‘ ๐‘›๐‘›ยฒ๐œƒ + ๐‘๐‘๐‘๐‘๐‘ ๐‘ ยฒ๐œƒ = 1

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Double Angle Formulas (sin 2x)

sin 2๐‘ฅ = 2 sin ๐‘ฅ cos ๐‘ฅ

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Double Angle Formula (cos 2x)

cos 2๐‘ฅ = cosยฒ๐‘ฅ โˆ’ sinยฒ๐‘ฅ = 1 โˆ’ 2 sinยฒ๐‘ฅ

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Power-Reducing Formulas (cosยฒx)

cosยฒ๐‘ฅ = (1/2) (1 + cos 2๐‘ฅ)

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Power-Reducing Formulas (sinยฒx)

sinยฒ๐‘ฅ = (1/2) (1 โˆ’ cos 2๐‘ฅ)

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Area Between Two Curves

๐‘†lices โŠฅ to x-axis: ๐ด = โˆซ [๐‘“(๐‘ฅ) โˆ’ ๐‘”(๐‘ฅ)] ๐‘‘๐‘ฅ from a to b. Slices โŠฅ to y-axis: ๐ด = โˆซ [๐‘“(๐‘ฆ) โˆ’ ๐‘”(๐‘ฆ)] ๐‘‘๐‘ฆ from c to d

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Volume By Disk Method

About x-axis: ๐‘‰ = ๐œ‹ โˆซ [๐‘…(๐‘ฅ)]ยฒ ๐‘‘๐‘ฅ from a to b. About y-axis: ๐‘‰ = ๐œ‹ โˆซ [๐‘…(๐‘ฆ)]ยฒ ๐‘‘๐‘ฆ from c to d

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Volume By Washer Method

About x-axis: ๐‘‰ = ๐œ‹ โˆซ ([๐‘…(๐‘ฅ)]ยฒ โˆ’ [๐‘Ÿ(๐‘ฅ)]ยฒ) ๐‘‘๐‘ฅ from a to b. About y-axis: ๐‘‰ = ๐œ‹ โˆซ ([๐‘…(๐‘ฆ)]ยฒ โˆ’ [๐‘Ÿ(๐‘ฆ)]ยฒ) ๐‘‘๐‘ฆ from c to d

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Volume By Shell Method

About x-axis: ๐‘‰ = 2 ๐œ‹ โˆซ ๐‘ฆ [๐‘…(๐‘ฆ)] ๐‘‘๐‘ฆ from c to d. About y-axis: ๐‘‰ = 2 ๐œ‹ โˆซ ๐‘ฅ [๐‘…(๐‘ฅ)] ๐‘‘๐‘ฅ from a to b

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General Equations for Known Cross Section

SQUARES: ๐‘‰ = โˆซ (๐‘๐‘Ž๐‘ ๐‘’)ยฒ ๐‘‘๐‘ฅ from a to b. TRIANGLES EQUILATERAL: ๐‘‰ = (โˆš3 / 4) โˆซ (๐‘๐‘Ž๐‘ ๐‘’)ยฒ ๐‘‘๐‘ฅ from a to b. ISOSCELES RIGHT: ๐‘‰ = (1 / 4) โˆซ (๐‘๐‘Ž๐‘ ๐‘’)ยฒ ๐‘‘๐‘ฅ from a to b. RECTANGLES: ๐‘‰ = โˆซ (๐‘๐‘Ž๐‘ ๐‘’) โˆ™ โ„Ž ๐‘‘๐‘ฅ from a to b where h is the height of the rectangles. SEMI-CIRCLES: ๐‘‰ = โˆซ (๐‘Ÿ๐‘Ž๐‘‘๐‘–๐‘ข๐‘ )ยฒ ๐‘‘๐‘ฅ from a to b where radius is ยฝ distance between the two curves.

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โˆซ ๐‘‘๐‘ข / ๐‘ข

ln| ๐‘ข | + ๐ถ

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โˆซ ๐‘ข^๐‘› ๐‘‘๐‘ข

๐‘ข^(๐‘›+1) / (๐‘› + 1) + ๐ถ (๐‘› โ‰  โˆ’1)

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โˆซ ๐‘’^๐‘ข ๐‘‘๐‘ข

๐‘’^๐‘ข + ๐ถ

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โˆซ ๐‘Ž^๐‘ข ๐‘‘๐‘ข

๐‘Ž^๐‘ข / ln ๐‘Ž + ๐ถ

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โˆซ sin ๐‘ข ๐‘‘๐‘ข

โˆ’ cos ๐‘ข + ๐ถ

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โˆซ cos ๐‘ข ๐‘‘๐‘ข

sin ๐‘ข + ๐ถ

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โˆซ tan ๐‘ข ๐‘‘๐‘ข

โˆ’ ln|cos ๐‘ข + ๐ถ|

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โˆซ cot ๐‘ข ๐‘‘๐‘ข

ln|sin ๐‘ข| + ๐ถ

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โˆซ sec ๐‘ข ๐‘‘๐‘ข

ln|sec ๐‘ข + tan ๐‘ข| + ๐ถ

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โˆซ csc ๐‘ข ๐‘‘๐‘ข

โˆ’ ln|csc ๐‘ข + cot ๐‘ข| + ๐ถ

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โˆซ secยฒ๐‘ข ๐‘‘๐‘‘๐‘‘๐‘‘

tan ๐‘ข + ๐ถ

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โˆซ cscยฒ๐‘ข ๐‘‘๐‘‘๐‘‘๐‘‘

โˆ’ cot ๐‘ข + ๐ถ

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โˆซ sec ๐‘ข tan ๐‘ข ๐‘‘๐‘‘๐‘‘๐‘‘

sec ๐‘ข + ๐ถ

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โˆซ csc ๐‘ข cot ๐‘ข ๐‘‘๐‘‘๐‘‘๐‘‘

โˆ’ csc ๐‘ข + ๐ถ

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๐‘‘/๐‘‘๐‘ฅ (sinโปยน(๐‘ข/๐‘Ž))

1 / โˆš(๐‘Žยฒโˆ’ ๐‘ขยฒ) * du/dx

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๐‘‘/๐‘‘๐‘ฅ [cotโปยน ๐‘ฅ]

-1 / (1 +xยฒ)

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๐‘‘/๐‘‘๐‘ฅ (secโปยน(๐‘ข/๐‘Ž))

๐‘Ž / (|๐‘ข|โˆš(๐‘ขยฒโˆ’ ๐‘Žยฒ)) * du/dx

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๐‘‘/๐‘‘๐‘ฅ (๐‘Ž^๐‘ข)

๐‘Ž^๐‘ข ln ๐‘Ž * du/dx

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๐‘‘/๐‘‘๐‘ฅ [logโ‚ ๐‘ฅ]

1 / (๐‘ฅ ln ๐‘Ž)

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โˆซ ๐‘‘๐‘ฅ / โˆš(๐‘Žยฒ โˆ’ ๐‘ขยฒ)

sinโปยน(๐‘ข/๐‘Ž) + ๐ถ