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Calculus Examples
Step 1
Step 1.1
Factor out of .
Step 1.2
Rewrite as exponentiation.
Step 2
Use the half-angle formula to rewrite as .
Step 3
Use the half-angle formula to rewrite as .
Step 4
Step 4.1
Let . Find .
Step 4.1.1
Differentiate .
Step 4.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3
Differentiate using the Power Rule which states that is where .
Step 4.1.4
Multiply by .
Step 4.2
Rewrite the problem using and .
Step 5
Step 5.1
Simplify.
Step 5.1.1
Multiply by .
Step 5.1.2
Multiply by .
Step 5.2
Simplify with commuting.
Step 5.2.1
Rewrite as a product.
Step 5.2.2
Rewrite as a product.
Step 5.3
Expand .
Step 5.3.1
Rewrite the exponentiation as a product.
Step 5.3.2
Apply the distributive property.
Step 5.3.3
Apply the distributive property.
Step 5.3.4
Apply the distributive property.
Step 5.3.5
Apply the distributive property.
Step 5.3.6
Apply the distributive property.
Step 5.3.7
Apply the distributive property.
Step 5.3.8
Apply the distributive property.
Step 5.3.9
Apply the distributive property.
Step 5.3.10
Apply the distributive property.
Step 5.3.11
Apply the distributive property.
Step 5.3.12
Apply the distributive property.
Step 5.3.13
Apply the distributive property.
Step 5.3.14
Apply the distributive property.
Step 5.3.15
Reorder and .
Step 5.3.16
Reorder and .
Step 5.3.17
Reorder and .
Step 5.3.18
Move .
Step 5.3.19
Move parentheses.
Step 5.3.20
Move parentheses.
Step 5.3.21
Move .
Step 5.3.22
Reorder and .
Step 5.3.23
Reorder and .
Step 5.3.24
Move parentheses.
Step 5.3.25
Move parentheses.
Step 5.3.26
Move .
Step 5.3.27
Reorder and .
Step 5.3.28
Reorder and .
Step 5.3.29
Move .
Step 5.3.30
Reorder and .
Step 5.3.31
Move parentheses.
Step 5.3.32
Move parentheses.
Step 5.3.33
Move .
Step 5.3.34
Reorder and .
Step 5.3.35
Move parentheses.
Step 5.3.36
Move parentheses.
Step 5.3.37
Reorder and .
Step 5.3.38
Reorder and .
Step 5.3.39
Reorder and .
Step 5.3.40
Move .
Step 5.3.41
Move parentheses.
Step 5.3.42
Move parentheses.
Step 5.3.43
Move .
Step 5.3.44
Move .
Step 5.3.45
Reorder and .
Step 5.3.46
Reorder and .
Step 5.3.47
Move parentheses.
Step 5.3.48
Move parentheses.
Step 5.3.49
Move .
Step 5.3.50
Move .
Step 5.3.51
Reorder and .
Step 5.3.52
Reorder and .
Step 5.3.53
Move .
Step 5.3.54
Reorder and .
Step 5.3.55
Move parentheses.
Step 5.3.56
Move parentheses.
Step 5.3.57
Move .
Step 5.3.58
Move .
Step 5.3.59
Reorder and .
Step 5.3.60
Move parentheses.
Step 5.3.61
Move parentheses.
Step 5.3.62
Multiply by .
Step 5.3.63
Multiply by .
Step 5.3.64
Multiply by .
Step 5.3.65
Multiply by .
Step 5.3.66
Multiply by .
Step 5.3.67
Multiply by .
Step 5.3.68
Multiply by .
Step 5.3.69
Multiply by .
Step 5.3.70
Multiply by .
Step 5.3.71
Multiply by .
Step 5.3.72
Multiply by .
Step 5.3.73
Multiply by .
Step 5.3.74
Multiply by .
Step 5.3.75
Combine and .
Step 5.3.76
Multiply by .
Step 5.3.77
Multiply by .
Step 5.3.78
Multiply by .
Step 5.3.79
Multiply by .
Step 5.3.80
Combine and .
Step 5.3.81
Multiply by .
Step 5.3.82
Multiply by .
Step 5.3.83
Multiply by .
Step 5.3.84
Multiply by .
Step 5.3.85
Multiply by .
Step 5.3.86
Combine and .
Step 5.3.87
Multiply by .
Step 5.3.88
Multiply by .
Step 5.3.89
Combine and .
Step 5.3.90
Raise to the power of .
Step 5.3.91
Raise to the power of .
Step 5.3.92
Use the power rule to combine exponents.
Step 5.3.93
Add and .
Step 5.3.94
Add and .
Step 5.3.95
Combine and .
Step 5.3.96
Multiply by .
Step 5.3.97
Multiply by .
Step 5.3.98
Combine and .
Step 5.3.99
Combine and .
Step 5.3.100
Multiply by .
Step 5.3.101
Combine and .
Step 5.3.102
Multiply by .
Step 5.3.103
Multiply by .
Step 5.3.104
Combine and .
Step 5.3.105
Combine and .
Step 5.3.106
Multiply by .
Step 5.3.107
Combine and .
Step 5.3.108
Multiply by .
Step 5.3.109
Combine and .
Step 5.3.110
Raise to the power of .
Step 5.3.111
Raise to the power of .
Step 5.3.112
Use the power rule to combine exponents.
Step 5.3.113
Add and .
Step 5.3.114
Multiply by .
Step 5.3.115
Combine and .
Step 5.3.116
Combine and .
Step 5.3.117
Multiply by .
Step 5.3.118
Combine and .
Step 5.3.119
Raise to the power of .
Step 5.3.120
Raise to the power of .
Step 5.3.121
Use the power rule to combine exponents.
Step 5.3.122
Add and .
Step 5.3.123
Combine and .
Step 5.3.124
Multiply by .
Step 5.3.125
Combine and .
Step 5.3.126
Combine and .
Step 5.3.127
Combine and .
Step 5.3.128
Combine and .
Step 5.3.129
Raise to the power of .
Step 5.3.130
Raise to the power of .
Step 5.3.131
Use the power rule to combine exponents.
Step 5.3.132
Add and .
Step 5.3.133
Multiply by .
Step 5.3.134
Multiply by .
Step 5.3.135
Combine and .
Step 5.3.136
Combine and .
Step 5.3.137
Raise to the power of .
Step 5.3.138
Use the power rule to combine exponents.
Step 5.3.139
Add and .
Step 5.3.140
Subtract from .
Step 5.3.141
Combine and .
Step 5.3.142
Reorder and .
Step 5.3.143
Reorder and .
Step 5.3.144
Reorder and .
Step 5.3.145
Move .
Step 5.3.146
Move .
Step 5.3.147
Move .
Step 5.3.148
Reorder and .
Step 5.3.149
Combine the numerators over the common denominator.
Step 5.3.150
Subtract from .
Step 5.3.151
Combine the numerators over the common denominator.
Step 5.3.152
Subtract from .
Step 5.4
Simplify.
Step 5.4.1
Rewrite as .
Step 5.4.2
Rewrite as a product.
Step 5.4.3
Multiply by .
Step 5.4.4
Multiply by .
Step 5.4.5
Move the negative in front of the fraction.
Step 6
Split the single integral into multiple integrals.
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Factor out .
Step 10
Using the Pythagorean Identity, rewrite as .
Step 11
Step 11.1
Let . Find .
Step 11.1.1
Differentiate .
Step 11.1.2
The derivative of with respect to is .
Step 11.2
Rewrite the problem using and .
Step 12
Split the single integral into multiple integrals.
Step 13
Apply the constant rule.
Step 14
Since is constant with respect to , move out of the integral.
Step 15
By the Power Rule, the integral of with respect to is .
Step 16
Combine and .
Step 17
Since is constant with respect to , move out of the integral.
Step 18
Since is constant with respect to , move out of the integral.
Step 19
Use the half-angle formula to rewrite as .
Step 20
Since is constant with respect to , move out of the integral.
Step 21
Step 21.1
Multiply by .
Step 21.2
Multiply by .
Step 22
Split the single integral into multiple integrals.
Step 23
Apply the constant rule.
Step 24
Step 24.1
Let . Find .
Step 24.1.1
Differentiate .
Step 24.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 24.1.3
Differentiate using the Power Rule which states that is where .
Step 24.1.4
Multiply by .
Step 24.2
Rewrite the problem using and .
Step 25
Combine and .
Step 26
Since is constant with respect to , move out of the integral.
Step 27
The integral of with respect to is .
Step 28
Apply the constant rule.
Step 29
Since is constant with respect to , move out of the integral.
Step 30
The integral of with respect to is .
Step 31
Step 31.1
Simplify.
Step 31.2
Simplify.
Step 31.2.1
To write as a fraction with a common denominator, multiply by .
Step 31.2.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 31.2.2.1
Multiply by .
Step 31.2.2.2
Multiply by .
Step 31.2.3
Combine the numerators over the common denominator.
Step 31.2.4
Move to the left of .
Step 31.2.5
Add and .
Step 32
Step 32.1
Replace all occurrences of with .
Step 32.2
Replace all occurrences of with .
Step 32.3
Replace all occurrences of with .
Step 32.4
Replace all occurrences of with .
Step 33
Step 33.1
Reduce the expression by cancelling the common factors.
Step 33.1.1
Factor out of .
Step 33.1.2
Factor out of .
Step 33.1.3
Cancel the common factor.
Step 33.1.4
Rewrite the expression.
Step 33.2
Multiply by .
Step 34
Reorder terms.