Integrals

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98 Terms

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∫dx

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∫uⁿ du

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∫e^u du

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∫b^u*ln(b) du

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∫b^u du

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∫cosu du

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∫-sinu du

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∫sinu du

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∫sec²u du

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∫-cscu*cotu du

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∫cscu*cotu du

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∫secu*tanu du

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∫-csc²u du

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∫csc²u du

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∫du/√(1-u²)

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∫-du/√(1-u²)

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∫du/(1+u²)

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∫-du/u√(1-u²)

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∫du/u√(1-u²)

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∫-du/(1+u²)

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∫du/u

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∫tan u du

-ln|cos u|+C

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∫cot u du

ln|sin u|+C

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∫sec u du

ln|sec u+tan u|+C

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∫csc u du

-ln|csc u+ cot u|+C

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f(x)g'(x) + g(x)f'(x)

derivative of the multiplaction of two functions

<p>derivative of the multiplaction of two functions</p>
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d/dx [f(x)/g(x)] =

quotient rule

<p>quotient rule</p>
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f'(g(x))g'(x)

chain rule

<p>chain rule</p>
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d/dx [ln|x|] =

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d/dx [a^x] =

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d/dx [log_a X] =

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∫- sinx

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∫secxtanx

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∫- cscxcotx

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d/dx [tanx]

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d/dx [cotx]

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∫cosx

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quotient rule

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derivative of a power

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chain rule

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derivative of an exponential function

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derivative of a natural log

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d/dx [a^x]

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derivative of a log base a

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d/dx [sinx]

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d/dx [cscx]

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d/dx [cosx]

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d/dx [secx]

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d/dx [tanx]

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d/dx [cotx]

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d/dx [arcsinx]

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d/dx [arccscx]

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d/dx [arccosx]

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d/dx [arcsecx]

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d/dx [arctanx]

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d/dx [arccotx]

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product rule

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f is continuous at x=c if…

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Intermediate Value Theorem

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

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Global Definition of a Derivative

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Alternative Definition of a Derivative

f '(x) is the limit of the following difference quotient as x approaches c

<p>f '(x) is the limit of the following difference quotient as x approaches c</p>
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∫-sin(x)

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∫sec²(x)

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∫-csc²(x)

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∫sec(x)tan(x)

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Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

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Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

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Mean Value Theorem

The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.

<p>The instantaneous rate of change will equal the mean rate of change somewhere in the interval. Or, the tangent line will be parallel to the secant line.</p>
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First Derivative Test for local extrema

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Point of inflection at x=k

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Combo Test for local extrema

If f'(c) = 0 and f"(c)

<p>If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c.
If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.</p>
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Horizontal Asymptote

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L'Hopital's Rule

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x+c

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sin(x)+C

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-cos(x)+C

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tan(x)+C

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-cot(x)+C

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sec(x)+C

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-csc(x)+C

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Fundamental Theorem of Calculus #1

The definite integral of a rate of change is the total change in the original function.

<p>The definite integral of a rate of change is the total change in the original function.</p>
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Fundamental Theorem of Calculus #2

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Mean Value Theorem for integrals or the average value of a functions

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ln(x)+C

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-ln(cosx)+C = ln(secx)+C

hint: tanu = sinu/cosu

<p>hint: tanu = sinu/cosu</p>
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ln(sinx)+C = -ln(cscx)+C

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ln(secx+tanx)+C = -ln(secx-tanx)+C

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ln(cscx+cotx)+C = -ln(cscx-cotx)+C

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If f and g are inverses of each other, g'(x)

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Exponential growth (use N= )

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Formula for Disk Method

Axis of rotation is a boundary of the region.

<p>Axis of rotation is a boundary of the region.</p>
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Formula for Washer Method

Axis of rotation is not a boundary of the region.

<p>Axis of rotation is not a boundary of the region.</p>
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Inverse Secant Antiderivative

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Inverse Sine Antiderivative

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Derivative of eⁿ

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ln(a)*aⁿ+C

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Derivative of ln(u)

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Antiderivative of f(x) from [a,b]

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