WolfreeAlpha

Free Wolfram Alpha Step-by-Step Solutions

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• Unlock FREE access to full Wolfram|Alpha Step-by-Step Solutions.
• Facilitate the use of Wolfram|Alpha Show Steps API.

Contributing

• Start a discussion to ask questions and collaborate with maintainers.
• Create an issue to request new features or report bugs.
• Open a pull request to submit a patch.

Branching

• main is the unstable development branch for daily builds.
• release is the stable production branch for deployable builds.

Workflow

• Every pull request spins up a disposable Deploy Preview.
• Maintainers merge main into release once it has acquired enough functionalities or fixes.
• Github then deploys release to http://WolfreeAlpha.github.io automagically.

License

• MIT License

Examples

Two-Dimensional Expression Input:
lim_n->inf (1+i/n)^n = int_-inf^i e^tdt = sum_n=0^inf i^n/n! = e^sqr-1
Subscripts are typeset with the underscore, while superscripts are made with the caret.

Differential Equations

See how separable equations are solved:
solve y' = y^2 x
y'(x) = (x + 2) e^(-y(x)), y(0) = 0
sec(y(t)) y'(t) + sin(t - y(t)) = sin(t + y(t))
Solve first-order linear equations:
y'(t) - 2y(t) = 3 e^(2t)
x y'(x) - 4 y(x) = x^6 exp(x), y(1) = 0
See steps that use Laplace transforms to solve an ODE:
solve y'(t) - 3y(t) = delta(t - 2), where y(0) = 0
Solve exact equations:
(3x + 2y)y' + 2x + 3y = 0 where y(0) = 2
solve t + arctan(y(t)) + (t + y(t))/(1 + y(t)^2) y'(t) = 0
Transform into an exact equation:
2 t exp(2y)y' = 3 t^4 + exp(2y)
Learn to solve Bernoulli equations:
y'(x) - y = e^x y^2
x'(t) = x(t)(t x(t)^3 - 1)
Apply a linear substitution:
v' = t sin(2v + t) - 1/2, v(0) = pi/2
Solve a first-order homogeneous equation through a substitution:
solve x y' = y*(log(x) - log(y))
Make general substitutions:
solve 2 t^3 y'(t) = 1 + sqrt(1 + 4 t^2 y(t))
y'(x) = (1-x cos(y(x))) cot(y(x))
Solve a Riccati equation:
x^2 v'(x) + 2 x v(x) = x^4 v(x)^2 + 4
solve y' = y^2/x^2 - y/x + 1, y(1) = 0
Solve an Abel equation of the first kind with a constant invariant:
y'(x) = e^(2x) x y(x)^3 - y(x) - x e^(-x), y(0) = 0
Solve a Chini equation with a constant invariant:
2 x'(t) + t = 4sqrt(x(t))
See the steps for solving Clairaut's equation:
y(x) = x y'(x) + y'(x)^2
Solve d'Alembert's equation:
x(t) = t x'(t)^2 + x'(t)
See how first-order ordinary differential equations are solved:
solve y' = 2((y + 2)/(x + y - 1))^2, y(1) = 0
t y(t) (1 + t y(t)^2) y'(t) = 1
Solve a second-order constant-coefficient linear homogeneous equation:
x''(t) = -k x(t)
solve y''(t) + 5y'(t) + 6y(t) = 0, y(0) = 1, y'(0) = 0
Solve a constant-coefficient linear equation with multiple methods:
solve y''+ y = sin(2x)
x'' - 2x' - 8x = 3e^(-2t), x(0) = 0, x'(0) = 1
See steps that use Laplace transforms to solve an ODE:
y''(t) + 2 y'(t) + 2 y(t) = cos(t) delta(t - 3 pi), y(0) = 1, y'(0) = -1
Reduce to a first-order equation:
t x''(t) - 2 x'(t) = 10 t^4
y''(x) + y'(x)^2 = 0
Derive the equation of a catenary curve:
solve v''(x)^2 = (1+v'(x)^2), v(0) = 1, v'(0) = 0
Solve Euler-Cauchy equations:
solve x^2 y''(x) - x y'(x) + y(x) = 0
x^2 y'' - y = 0
2t^2*y'' + t*y' - 3*y = t, y(1) = 0, y'(1) = 1
See how general second-order ordinary differential equations are solved:
t y''(t) - t y'(t) + y(t) = 2, y(0) = 2, y'(0) = -4
solve y''(t) + sin(y(t)) = 0
y'' - 2 cot(x) y' + (1+2cot(x)^2) y = 0
y''(x) + tan(x) y'(x) + sec(x)^2 y(x)==0
x^4*y*y" + x^4*y'*y' + 3*x^3*y*y' = 1
x^2y'' + xy' + (x^2-1/4)y=0
See the steps for higher-order equations:
solve y''''(x) + 16y(x) = 0
y''' - 2y'' + y' = 2 - 24e^t + 40e^(5t), y(0) = 1, y'(0) = 0, y''(0) = -1
y''' - y'' + y' - y = cosh(x)
y''''''(t) - 4y'''''(t) + 7y''''(t) - 4y'''(t) - 4y''(t) + 8y'(t) - 4y(t) = 0

Calculus

Find the derivative using the product rule, chain rule and other methods:
d/dx sqrt(x)
derivative of x^4 + 9x^3 + 7x - 2
slope of log(t) / (t^2 + 1)
Find partial derivatives of multivariate expressions:
partial derivative of exp(-x^2 - y^2)
partial derivative of cos( 3 pi /4 * x + 5 pi /3 * y + pi / 2 * z) with respect to y
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))
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
Learn how to take a limit:
limit of (x - 3) / (x^2 - 2x - 3) as x approaches 3
take the limit as x goes to infinity: (1 + 1/x)^x
limit t*tan(t) as t->pi/2 from the left
lim (x^2 + 2x + 3)/(x^2 - 2x - 3) as x->3+
Find local and global extrema using a variety of tests:
local extrema of x^3 - 3x + 1
find the extrema of 3x^4 - 4x^3 - 12x^2 + 1
maximize sin(x)
Find a tangent line to a curve using the derivative at a point:
tangent line to sin(x) - 1 at x=0
tangent line of y=x^2 at x=1
Calculate a multivariate tangent plane by examining partial derivatives:
tangent plane to z=x^2+y^2 at x=1,y=1
Get hints for discontinuities:
where is (x^2 - 4)/(x^2 - 2x - 3) discontinuous?
discontinuities of sin((x^2-x)/(x-1)) + log(sqrt(abs(x)))
Calculate inflection points step by step:
inflection points x^3 - 3 x^2 + 1
points of inflection t * cbrt(t^2 - 4)
inflection points of 3sin(2x)
inflection pts sgn(x)|x|^(3/2) + x^3 - 3x^2 + 1 over -2 < x < 3
Learn how to compute the area between curves:
area between 1/2x(x-2)(x+2) and x
area of the region between y=x and y=x^2 for -1/2<x<1
Calculate the arc length of a curve step by step:
arc length of Sin[x] for x between 0 and 2pi
Specify the curve in polar coordinates:
arc length of r=theta^2 for 0<theta<2pi
Specify the curve parametrically:
arc length of {Sin[x], Cos[x], x, x} for 0<x<3pi

Proofs

See the steps toward proving a trigonometric identity:
(1 + tan(x))/(1 - tan(x)) = (cos(x) + sin(x))/(cos(x) - sin(x))
Prove a sum identity by induction:
prove by induction sum of j from 1 to n = n(n+1)/2 for n>0
Prove divisibility by induction:
using induction, prove 9^n-1 is divisible by 4 assuming n>0
Prove an inequality by induction:
prove by induction (3n)! > 3^n (n!)^3 for n>0

Algebra

Solve equations one step at a time:
x^2 + 5x + 6 = 0
Factor polynomials step by step:
factor x^3+2x^2+x+2
Expand polynomials using FOIL, the binomial theorem and other methods:
expanded form of (x + 3)^2
Learn to rewrite a rational function using our step-by-step partial fraction decomposition:
partial fraction decomposition 1/(x^2 + 4x + 3)

Discrete Mathematics

Find step-by-step solutions for prime factorization, primality testing, GCD and more:
is 73 prime?
lcm(90, 342)
are 12 and 21 coprime?
Follow the steps to convert between bases:
convert 33ab3653 base 12 to base 10

Linear Algebra

Compute properties of a matrix step by step:
{{2,4,4},{1,2,2},{0,7,3}}
Find the determinant step by step with various methods:
determinant of {{1,2,1}, {1,1,0}, {0,1,1}}
Compute eigenvalues and eigenvectors step by step:
eigenvalues {{3,-1},{0,2}}
Compute a cross product step by step:
(1,2,3)x(3,4,5)

Geometry

Compute properties of geometric objects step by step:
area of a circle with radius 2
Learn details of some geometric computations:
distance between (3, 2) and (5, 1)
slope of line through (2, 3) and (4, -1)
Determine the equation of a line with certain properties using a selected form:
equation of line through (2, 5), (4, 1)

Statistics

Perform basic statistical analysis on a set of data, seeing the steps along the way:
average of {1, 2, 3, 4, 5, 6, 7}
mode of {1, 5, 4, 2, 3, 4, 5, 11, 4, 11, 20}
variance 4, 6, 7, 9, 10, 11
quartile of {1, 2, 3, 4, 5, 6}

Chemistry

Get a step-by-step procedure to draw the Lewis structures of molecules:
What is the Lewis structure of NO2
Follow the steps for unit conversion:
convert 5mL benzene to moles
Learn how to balance a chemical reaction:
glucose + oxygen -> water + carbon dioxide
Compute oxidation states of chemicals:
oxidation states of chromium(III) chloride

Physics Formulas

Compute mechanical work:
work F=30N, d=100m
Compute centripetal acceleration:
centripetal acceleration, 30mph, 500 ft
Compute a Doppler shift:
Doppler shift 300Hz, 75mph
Do computations using the thin lens equation:
lens, f=20ft, o=400ft
Compute a diffraction pattern for a circular aperture:
circular aperture a=0.2mm, lambda=650nm
Compute a diffraction pattern for a single slit:
single slit diffraction d=1/16 inch, lambda=200nm
Do relativistic addition of velocities:
add velocities, 8e7 m/s, 8e7 m/s
Compute relativistic momentum:
relativistic momentum electron, 0.8c
Determine gravitational time dilation near massive objects:
gravitational time dilation at 1000 km and 100 solar masses
Compute the orbital period of a massive body orbiting another:
Kepler's third law, 4 solar masses, 5 Earth masses, 2.5 AU
Calculate flow through a pipe:
water flow through a pipe 4" diameter