Calculus Examples

Evaluate the Integral integral of sin(x)^6 with respect to x
Step 1
Simplify with factoring out.
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Step 1.1
Factor out of .
Step 1.2
Rewrite as exponentiation.
Step 2
Use the half-angle formula to rewrite as .
Step 3
Let . Then , so . Rewrite using and .
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Step 3.1
Let . Find .
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Step 3.1.1
Differentiate .
Step 3.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3
Differentiate using the Power Rule which states that is where .
Step 3.1.4
Multiply by .
Step 3.2
Rewrite the problem using and .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Simplify by multiplying through.
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Step 5.1
Rewrite as a product.
Step 5.2
Expand .
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Step 5.2.1
Rewrite the exponentiation as a product.
Step 5.2.2
Rewrite the exponentiation as a product.
Step 5.2.3
Apply the distributive property.
Step 5.2.4
Apply the distributive property.
Step 5.2.5
Apply the distributive property.
Step 5.2.6
Apply the distributive property.
Step 5.2.7
Apply the distributive property.
Step 5.2.8
Apply the distributive property.
Step 5.2.9
Apply the distributive property.
Step 5.2.10
Apply the distributive property.
Step 5.2.11
Apply the distributive property.
Step 5.2.12
Apply the distributive property.
Step 5.2.13
Apply the distributive property.
Step 5.2.14
Apply the distributive property.
Step 5.2.15
Apply the distributive property.
Step 5.2.16
Reorder and .
Step 5.2.17
Reorder and .
Step 5.2.18
Reorder and .
Step 5.2.19
Move .
Step 5.2.20
Move parentheses.
Step 5.2.21
Move parentheses.
Step 5.2.22
Move .
Step 5.2.23
Reorder and .
Step 5.2.24
Reorder and .
Step 5.2.25
Reorder and .
Step 5.2.26
Move parentheses.
Step 5.2.27
Move .
Step 5.2.28
Move parentheses.
Step 5.2.29
Move parentheses.
Step 5.2.30
Move parentheses.
Step 5.2.31
Move .
Step 5.2.32
Reorder and .
Step 5.2.33
Reorder and .
Step 5.2.34
Reorder and .
Step 5.2.35
Move .
Step 5.2.36
Move .
Step 5.2.37
Move parentheses.
Step 5.2.38
Move parentheses.
Step 5.2.39
Move parentheses.
Step 5.2.40
Move .
Step 5.2.41
Reorder and .
Step 5.2.42
Reorder and .
Step 5.2.43
Reorder and .
Step 5.2.44
Move parentheses.
Step 5.2.45
Move .
Step 5.2.46
Move .
Step 5.2.47
Move parentheses.
Step 5.2.48
Move parentheses.
Step 5.2.49
Move parentheses.
Step 5.2.50
Move parentheses.
Step 5.2.51
Move .
Step 5.2.52
Reorder and .
Step 5.2.53
Reorder and .
Step 5.2.54
Reorder and .
Step 5.2.55
Move .
Step 5.2.56
Move parentheses.
Step 5.2.57
Move parentheses.
Step 5.2.58
Move .
Step 5.2.59
Move .
Step 5.2.60
Reorder and .
Step 5.2.61
Reorder and .
Step 5.2.62
Reorder and .
Step 5.2.63
Move parentheses.
Step 5.2.64
Move .
Step 5.2.65
Move parentheses.
Step 5.2.66
Move parentheses.
Step 5.2.67
Move parentheses.
Step 5.2.68
Move .
Step 5.2.69
Move .
Step 5.2.70
Reorder and .
Step 5.2.71
Reorder and .
Step 5.2.72
Reorder and .
Step 5.2.73
Move .
Step 5.2.74
Move .
Step 5.2.75
Move parentheses.
Step 5.2.76
Move parentheses.
Step 5.2.77
Move parentheses.
Step 5.2.78
Move .
Step 5.2.79
Move .
Step 5.2.80
Reorder and .
Step 5.2.81
Reorder and .
Step 5.2.82
Reorder and .
Step 5.2.83
Move parentheses.
Step 5.2.84
Move .
Step 5.2.85
Move .
Step 5.2.86
Move parentheses.
Step 5.2.87
Move parentheses.
Step 5.2.88
Move parentheses.
Step 5.2.89
Move parentheses.
Step 5.2.90
Move .
Step 5.2.91
Move .
Step 5.2.92
Multiply by .
Step 5.2.93
Multiply by .
Step 5.2.94
Multiply by .
Step 5.2.95
Multiply by .
Step 5.2.96
Multiply by .
Step 5.2.97
Multiply by .
Step 5.2.98
Multiply by .
Step 5.2.99
Multiply by .
Step 5.2.100
Multiply by .
Step 5.2.101
Combine and .
Step 5.2.102
Multiply by .
Step 5.2.103
Combine and .
Step 5.2.104
Multiply by .
Step 5.2.105
Combine and .
Step 5.2.106
Multiply by .
Step 5.2.107
Multiply by .
Step 5.2.108
Combine and .
Step 5.2.109
Multiply by .
Step 5.2.110
Combine and .
Step 5.2.111
Combine and .
Step 5.2.112
Multiply by .
Step 5.2.113
Multiply by .
Step 5.2.114
Multiply by .
Step 5.2.115
Multiply by .
Step 5.2.116
Multiply by .
Step 5.2.117
Multiply by .
Step 5.2.118
Combine and .
Step 5.2.119
Multiply by .
Step 5.2.120
Multiply by .
Step 5.2.121
Combine and .
Step 5.2.122
Raise to the power of .
Step 5.2.123
Raise to the power of .
Step 5.2.124
Use the power rule to combine exponents.
Step 5.2.125
Add and .
Step 5.2.126
Subtract from .
Step 5.2.127
Combine and .
Step 5.2.128
Multiply by .
Step 5.2.129
Multiply by .
Step 5.2.130
Combine and .
Step 5.2.131
Combine and .
Step 5.2.132
Multiply by .
Step 5.2.133
Combine and .
Step 5.2.134
Multiply by .
Step 5.2.135
Multiply by .
Step 5.2.136
Multiply by .
Step 5.2.137
Multiply by .
Step 5.2.138
Combine and .
Step 5.2.139
Multiply by .
Step 5.2.140
Multiply by .
Step 5.2.141
Multiply by .
Step 5.2.142
Multiply by .
Step 5.2.143
Combine and .
Step 5.2.144
Raise to the power of .
Step 5.2.145
Raise to the power of .
Step 5.2.146
Use the power rule to combine exponents.
Step 5.2.147
Add and .
Step 5.2.148
Multiply by .
Step 5.2.149
Multiply by .
Step 5.2.150
Multiply by .
Step 5.2.151
Combine and .
Step 5.2.152
Multiply by .
Step 5.2.153
Multiply by .
Step 5.2.154
Combine and .
Step 5.2.155
Raise to the power of .
Step 5.2.156
Raise to the power of .
Step 5.2.157
Use the power rule to combine exponents.
Step 5.2.158
Add and .
Step 5.2.159
Multiply by .
Step 5.2.160
Multiply by .
Step 5.2.161
Multiply by .
Step 5.2.162
Multiply by .
Step 5.2.163
Combine and .
Step 5.2.164
Combine and .
Step 5.2.165
Multiply by .
Step 5.2.166
Combine and .
Step 5.2.167
Raise to the power of .
Step 5.2.168
Raise to the power of .
Step 5.2.169
Use the power rule to combine exponents.
Step 5.2.170
Add and .
Step 5.2.171
Combine and .
Step 5.2.172
Multiply by .
Step 5.2.173
Combine and .
Step 5.2.174
Raise to the power of .
Step 5.2.175
Use the power rule to combine exponents.
Step 5.2.176
Add and .
Step 5.2.177
Add and .
Step 5.2.178
Combine and .
Step 5.2.179
Reorder and .
Step 5.2.180
Reorder and .
Step 5.2.181
Reorder and .
Step 5.2.182
Move .
Step 5.2.183
Move .
Step 5.2.184
Move .
Step 5.2.185
Reorder and .
Step 5.2.186
Combine the numerators over the common denominator.
Step 5.2.187
Add and .
Step 5.2.188
Combine the numerators over the common denominator.
Step 5.2.189
Subtract from .
Step 5.3
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
Let . Then , so . Rewrite using and .
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Step 11.1
Let . Find .
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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
Use the half-angle formula to rewrite as .
Step 19
Since is constant with respect to , move out of the integral.
Step 20
Simplify.
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Step 20.1
Multiply by .
Step 20.2
Multiply by .
Step 21
Split the single integral into multiple integrals.
Step 22
Apply the constant rule.
Step 23
Let . Then , so . Rewrite using and .
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Step 23.1
Let . Find .
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Step 23.1.1
Differentiate .
Step 23.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 23.1.3
Differentiate using the Power Rule which states that is where .
Step 23.1.4
Multiply by .
Step 23.2
Rewrite the problem using and .
Step 24
Combine and .
Step 25
Since is constant with respect to , move out of the integral.
Step 26
The integral of with respect to is .
Step 27
Apply the constant rule.
Step 28
Combine and .
Step 29
Since is constant with respect to , move out of the integral.
Step 30
Since is constant with respect to , move out of the integral.
Step 31
The integral of with respect to is .
Step 32
Simplify.
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Step 32.1
Simplify.
Step 32.2
Simplify.
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Step 32.2.1
To write as a fraction with a common denominator, multiply by .
Step 32.2.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 32.2.2.1
Multiply by .
Step 32.2.2.2
Multiply by .
Step 32.2.3
Combine the numerators over the common denominator.
Step 32.2.4
Move to the left of .
Step 32.2.5
Add and .
Step 33
Substitute back in for each integration substitution variable.
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Step 33.1
Replace all occurrences of with .
Step 33.2
Replace all occurrences of with .
Step 33.3
Replace all occurrences of with .
Step 33.4
Replace all occurrences of with .
Step 34
Simplify.
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Step 34.1
Combine the numerators over the common denominator.
Step 34.2
Subtract from .
Step 34.3
Simplify each term.
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Step 34.3.1
Cancel the common factor of and .
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Step 34.3.1.1
Factor out of .
Step 34.3.1.2
Cancel the common factors.
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Step 34.3.1.2.1
Factor out of .
Step 34.3.1.2.2
Cancel the common factor.
Step 34.3.1.2.3
Rewrite the expression.
Step 34.3.2
Move the negative in front of the fraction.
Step 34.3.3
Cancel the common factor of and .
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Step 34.3.3.1
Factor out of .
Step 34.3.3.2
Cancel the common factors.
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Step 34.3.3.2.1
Factor out of .
Step 34.3.3.2.2
Cancel the common factor.
Step 34.3.3.2.3
Rewrite the expression.
Step 34.3.4
Multiply by .
Step 34.4
Apply the distributive property.
Step 34.5
Simplify.
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Step 34.5.1
Multiply .
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Step 34.5.1.1
Multiply by .
Step 34.5.1.2
Multiply by .
Step 34.5.2
Combine.
Step 34.5.3
Multiply .
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Step 34.5.3.1
Multiply by .
Step 34.5.3.2
Multiply by .
Step 34.5.4
Multiply .
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Step 34.5.4.1
Multiply by .
Step 34.5.4.2
Multiply by .
Step 34.6
Simplify each term.
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Step 34.6.1
Multiply by .
Step 34.6.2
Multiply by .
Step 35
Reorder terms.