Wolfram|Alpha Examples: Step-by-Step Calculus

Limits

Learn how to take a limit:

limit of (x - 3) / (x^2 - 2x - 3) as x approaches 3

Calculate a limit approaching infinity:

take the limit as x goes to infinity: (1 + 1/x)^x

Perform a one-sided limit step by step:

limit t*tan(t) as t->pi/2 from the left

lim (x^2 + 2x + 3)/(x^2 - 2x - 3) as x->3+

Use L'Hôpital's rule to compute a limit:

Limit of 5 Log[x]/(x - 1) as x->1

See the steps to use the squeeze theorem on a limit:

limit of sin[x]/x as x->0

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Derivatives

Explore the steps for derivative calculations using the power rule:

d/dx sqrt(x)

Find the derivative using the product rule:

compute the derivative of x*sin[x]

Compute derivatives using the quotient rule:

d/dx of (x^2-x)/(x^2+3)

See step by step how to apply the chain rule:

derivative of (cos(log(x)))^3

Perform logarithmic differentiation to compute a derivative:

first derivative of 1/x^3

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Integrals

Calculate an indefinite integral by substitution, integration by parts and other methods:

integrate sin(x)cos(x)^2

integral of sqrt(a^2 - x^2)

show the integration: arcsec(sqrt(t))

Use u -substitution to compute an integral step by step:

integrate x*cos(x^2+1)

integrate (2x^3+1)^7(x^2) for 0<x<1

Compute an integral by parts:

integrate by parts x*cos(x)

Integrate using trigonometric substitution:

integrate tan[x]

See the steps for definite integration over an interval:

integrate sin(t)^2cos(t) from 0 to pi/2

integrate 1/(x^2 + 1), x = -infinity ... infinity

integrate 1/(cos(x) + 2) from 0 to 2pi

See the steps for multivariate integration over a given domain:

integrate sin(x)cos(y), x = 0 to pi, y = 0 to pi/2

int sin^2 x + y sin z dx dy dz , x=0..pi, y=0..1, z=0..pi

integrate x^2 sin y dx dy, x=0..y, y=0..pi

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Continuity

Get step-by-step solutions for finding discontinuities:

where is (x^2 - 4)/(x^2 - 2x - 3) discontinuous?

discontinuities of sin((x^2-x)/(x-1)) + log(sqrt(abs(x)))

Determine the continuity of a function at a point:

(x-5)/(x-5) continuous for x = 5

is 3 Cos[5 x] continuous at x = 1?

is Piecewise[{{x^2, x < 0},{4x + 8, x >= 0}}] continuous at x = 0?

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Sum Convergence

Learn how to compute the convergence of an infinite sum:

sum convergence of n

does the sum of 2^(-n) converge

does the sum of 5*3^(1 - n) converge

Determine whether a series converges step by step:

does the sum of 1/n^2 converge?

convergence of the sum (2/3)^x

Test whether a series converges absolutely:

does the sum of (-2)^(-n) converge absolutely

Explore divergent series:

sum convergence of 1/n

sum convergence of u/(u^2+1) for u=1 to infinity

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Vector Analysis

See the steps to compute the gradient:

Calculate the gradient of (x - 4)*Log[x y]

del (x - 4)*(x - 2)*(y - z)

Learn how to compute the divergence:

div (x^2-y^2, 2xy)

divergence {x+y, x Sin[z], z+y/x^2}

Calculate the curl:

curl [3yz, 2xz, 5yx]

curl of {y^2 cos z, z sin y, e^x}

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Applications of Calculus

Find a line tangent to a curve using the derivative at a point:

tangent line to sin(x) - 1 at x=0

Calculate inflection points step by step:

inflection points x^3 - 3 x^2 + 1

Explore step by step a variety of tests to discover the extrema:

find the extrema of 3x^4 - 4x^3 - 12x^2 + 1

Find the average value of an equation on an interval:

average value of 1/x for 1<x<5

Learn how to compute the area between curves:

area between 1/2x(x-2)(x+2) and x

Calculate the arc length of a curve step by step:

arc length of Sin[x] for x between 0 and 2pi

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