File size: 6,298 Bytes
7008d22 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | # 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
```
|