Examples (generated by math-steps.jar)
diff x^2 sin(x)
Step 1: Product rule: (f g)' = f' g + f g', with f = x^2 and g = sin(x)
(f*g)' = f'*g + f*g', f = x^2, g = sin(x)
- Power rule: d/dx [x^n] = n x^(n-1)
d/dx [x^2] = 2x
- Standard derivative of sin(x)
d/dx [sin(x)] = cos(x)
Step 2: Put it together
d/dx [x^2*sin(x)] = x^2*cos(x) + 2x*sin(x)
Step 3: Answer
d/dx [x^2*sin(x)] = x^2*cos(x) + 2x*sin(x)
diff (x+1)/(x-1)
Step 1: Rewrite the expression
(x + 1)/(x - 1)
Step 2: Quotient rule: (u/v)' = (u'v - uv') / v^2, with u = x + 1 and v = x - 1
(u/v)' = (u'v - uv')/v^2, u = x + 1, v = x - 1
- Sum rule: differentiate each term separately
d/dx [x + 1] = d/dx [x] + d/dx [1]
- Derivative of x
d/dx [x] = 1
- Constant rule: the derivative of a constant is 0
d/dx [1] = 0
- Add the results
d/dx [x + 1] = 1
- Sum rule: differentiate each term separately
d/dx [x - 1] = d/dx [x] + d/dx [-1]
- Derivative of x
d/dx [x] = 1
- Constant rule: the derivative of a constant is 0
d/dx [-1] = 0
- Add the results
d/dx [x - 1] = 1
Step 3: Put it together and simplify
d/dx [(x + 1)/(x - 1)] = -2/(x - 1)^2
Step 4: Answer
d/dx [(x + 1)/(x - 1)] = -2/(x - 1)^2
diff e^(sin(3x))
Step 1: Chain rule: d/dx [e^u] = e^u * u', where u = sin(3x)
d/dx [e^u] = e^u*u', u = sin(3x)
- Chain rule: d/dx [sin(u)] = cos(u) * u', where u = 3x
d/dx [sin(u)] = cos(u) * u', u = 3x
- Constant multiple rule: d/dx [c x] = c
d/dx [3x] = 3
- Put it together
d/dx [sin(3x)] = 3cos(3x)
Step 2: Put it together
d/dx [e^(sin(3x))] = 3cos(3x)*e^(sin(3x))
Step 3: Answer
d/dx [e^(sin(3x))] = 3cos(3x)*e^(sin(3x))
int x e^x
Step 1: Integration by parts: integral u dv = u v - integral v du (choose u by LIATE: logs, inverse trig, algebraic, trig, exponential)
u = x, dv = e^x dx, du = 1 dx, v = e^x
Step 2: Now integrate v du
integral e^x dx
- Standard integral: integral e^u du = e^u
Standard integral: integral e^u du = e^u => integral e^x dx = e^x
Step 3: So u v - integral v du
integral x*e^x dx = x*e^x - e^x
Step 4: Answer (add the constant of integration C)
integral x*e^x dx = x*e^x - e^x + C
Step 5: Check: differentiating the answer gives back the integrand
d/dx [x*e^x - e^x] = x*e^x (checked)
int 2x cos(x^2)
Step 1: Constant multiple rule: take the constant 2 outside the integral
integral 2x*cos(x^2) dx = 2 * integral x*cos(x^2) dx
- Substitution: let u = x^2
u = x^2, du = 2x dx
- The integral becomes
integral cos(u)/2 du
- Constant multiple rule: take the constant 1/2 outside the integral
integral cos(u)/2 du = 1/2 * integral cos(u) du
- Standard integral: integral cos(u) du = sin(u)
integral cos(u) du = sin(u)
- Multiply by the constant
integral cos(u)/2 du = sin(u)/2
- Substitute back u = x^2
integral x*cos(x^2) dx = sin(x^2)/2
Step 2: Multiply by the constant
integral 2x*cos(x^2) dx = sin(x^2)
Step 3: Answer (add the constant of integration C)
integral 2x*cos(x^2) dx = sin(x^2) + C
Step 4: Check: differentiating the answer gives back the integrand
d/dx [sin(x^2)] = 2x*cos(x^2) (checked)
int (x+3)/(x^2+3x+2)
Step 1: Rewrite the expression
(x + 3)/(x^2 + 3x + 2)
Step 2: The integrand is a ratio of polynomials: use partial fractions
integral (x + 3)/(x^2 + 3x + 2) dx
Step 3: Partial fractions: factor the denominator as (x + 1)(x + 2) and split the fraction
(x + 3)/(x^2 + 3x + 2) = 2/(x + 1) - 1/(x + 2)
- integral A/(x - r) dx = A ln|x - r|
integral 2/(x + 1) dx = 2ln|x + 1|
- integral A/(x - r) dx = A ln|x - r|
integral -1/(x + 2) dx = -ln|x + 2|
Step 4: Combine
integral (x + 3)/(x^2 + 3x + 2) dx = 2ln|x + 1| - ln|x + 2|
Step 5: Answer (add the constant of integration C)
integral (x + 3)/(x^2 + 3x + 2) dx = 2ln|x + 1| - ln|x + 2| + C
Step 6: Check: differentiating the answer gives back the integrand
d/dx [2ln|x + 1| - ln|x + 2|] = (x + 3)/(x^2 + 3x + 2) (checked)
int sin(x)^2
Step 1: Use the identity sin^2(u) = (1 - cos(2u))/2
Use the identity sin^2(u) = (1 - cos(2u))/2 => integral sin(x)^2 dx = integral (1 - cos(2x))/2 dx
- Constant multiple rule: take the constant 1/2 outside the integral
integral (1 - cos(2x))/2 dx = 1/2 * integral (1 - cos(2x)) dx
- Sum rule: integrate each term separately
integral (1 - cos(2x)) dx = integral -cos(2x) dx + integral 1 dx
- Constant multiple rule: take the constant -1 outside the integral
integral -cos(2x) dx = -1 * integral cos(2x) dx
- Standard integral: integral cos(u) du = sin(u), with the linear substitution u = 2x (du = 2 dx, so divide by 2)
Standard integral: integral cos(u) du = sin(u), u = 2x => integral cos(2x) dx = sin(2x)/2
- Multiply by the constant
integral -cos(2x) dx = -sin(2x)/2
- Integral of a constant: integral c dx = c x
integral 1 dx = x
- Add the results
integral (1 - cos(2x)) dx = x - sin(2x)/2
- Multiply by the constant
integral (1 - cos(2x))/2 dx = (x - sin(2x)/2)/2
Step 2: So
integral sin(x)^2 dx = (x - sin(2x)/2)/2
Step 3: Answer (add the constant of integration C)
integral sin(x)^2 dx = x/2 - sin(2x)/4 + C
Step 4: Check: differentiating the answer gives back the integrand
d/dx [x/2 - sin(2x)/4] = sin(x)^2 (checked)
defint sin(x) from 0 to pi
Step 1: Standard integral: integral sin(u) du = -cos(u)
Standard integral: integral sin(u) du = -cos(u) => integral sin(x) dx = -cos(x)
Step 2: Fundamental theorem of calculus: integral from a to b of f = F(b) - F(a), with F = -cos(x)
integral from 0 to pi = [-cos(x)] from 0 to pi
Step 3: Evaluate at the limits
F(pi) - F(0) = (1) - (-1) = 2
Step 4: Answer
integral from 0 to pi of sin(x) dx = 2
Step 5: Check: a numerical integration gives the same value
Simpson's rule ~ 2
