Calculus Examples

Evaluate the Integral integral of sin(x)^4cos(x)^2 with respect to x
Step 1
Use the half-angle formula to rewrite as .
Step 2
Simplify with factoring out.
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Step 2.1
Factor out of .
Step 2.2
Rewrite as exponentiation.
Step 3
Use the half-angle formula to rewrite as .
Step 4
Let . Then , so . Rewrite using and .
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Step 4.1
Let . Find .
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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
Simplify by multiplying through.
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Step 5.1
Simplify.
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Step 5.1.1
Multiply by .
Step 5.1.2
Multiply by .
Step 5.2
Simplify with commuting.
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Step 5.2.1
Rewrite as a product.
Step 5.2.2
Rewrite as a product.
Step 5.3
Expand .
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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
Move .
Step 5.3.18
Reorder and .
Step 5.3.19
Move .
Step 5.3.20
Move .
Step 5.3.21
Reorder and .
Step 5.3.22
Reorder and .
Step 5.3.23
Move .
Step 5.3.24
Reorder and .
Step 5.3.25
Reorder and .
Step 5.3.26
Move parentheses.
Step 5.3.27
Move .
Step 5.3.28
Reorder and .
Step 5.3.29
Move .
Step 5.3.30
Move .
Step 5.3.31
Move .
Step 5.3.32
Reorder and .
Step 5.3.33
Reorder and .
Step 5.3.34
Move parentheses.
Step 5.3.35
Move .
Step 5.3.36
Reorder and .
Step 5.3.37
Reorder and .
Step 5.3.38
Move .
Step 5.3.39
Move .
Step 5.3.40
Reorder and .
Step 5.3.41
Move .
Step 5.3.42
Move .
Step 5.3.43
Move .
Step 5.3.44
Reorder and .
Step 5.3.45
Reorder and .
Step 5.3.46
Move .
Step 5.3.47
Move .
Step 5.3.48
Reorder and .
Step 5.3.49
Reorder and .
Step 5.3.50
Move parentheses.
Step 5.3.51
Move .
Step 5.3.52
Move .
Step 5.3.53
Reorder and .
Step 5.3.54
Move .
Step 5.3.55
Move .
Step 5.3.56
Move .
Step 5.3.57
Move .
Step 5.3.58
Reorder and .
Step 5.3.59
Reorder and .
Step 5.3.60
Move parentheses.
Step 5.3.61
Move .
Step 5.3.62
Move .
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
Multiply by .
Step 5.3.76
Combine and .
Step 5.3.77
Multiply by .
Step 5.3.78
Multiply by .
Step 5.3.79
Combine and .
Step 5.3.80
Multiply by .
Step 5.3.81
Combine and .
Step 5.3.82
Combine and .
Step 5.3.83
Multiply by .
Step 5.3.84
Multiply by .
Step 5.3.85
Combine and .
Step 5.3.86
Multiply by .
Step 5.3.87
Combine and .
Step 5.3.88
Combine and .
Step 5.3.89
Multiply by .
Step 5.3.90
Combine and .
Step 5.3.91
Raise to the power of .
Step 5.3.92
Raise to the power of .
Step 5.3.93
Use the power rule to combine exponents.
Step 5.3.94
Add and .
Step 5.3.95
Subtract from .
Step 5.3.96
Subtract from .
Step 5.3.97
Multiply by .
Step 5.3.98
Multiply by .
Step 5.3.99
Combine and .
Step 5.3.100
Combine and .
Step 5.3.101
Multiply by .
Step 5.3.102
Combine and .
Step 5.3.103
Multiply by .
Step 5.3.104
Multiply by .
Step 5.3.105
Combine and .
Step 5.3.106
Combine and .
Step 5.3.107
Multiply by .
Step 5.3.108
Combine and .
Step 5.3.109
Multiply by .
Step 5.3.110
Combine and .
Step 5.3.111
Raise to the power of .
Step 5.3.112
Raise to the power of .
Step 5.3.113
Use the power rule to combine exponents.
Step 5.3.114
Add and .
Step 5.3.115
Multiply by .
Step 5.3.116
Multiply by .
Step 5.3.117
Multiply by .
Step 5.3.118
Combine and .
Step 5.3.119
Multiply by .
Step 5.3.120
Multiply by .
Step 5.3.121
Combine and .
Step 5.3.122
Raise to the power of .
Step 5.3.123
Raise to the power of .
Step 5.3.124
Use the power rule to combine exponents.
Step 5.3.125
Add and .
Step 5.3.126
Multiply by .
Step 5.3.127
Multiply by .
Step 5.3.128
Multiply by .
Step 5.3.129
Multiply by .
Step 5.3.130
Combine and .
Step 5.3.131
Multiply by .
Step 5.3.132
Multiply by .
Step 5.3.133
Combine and .
Step 5.3.134
Raise to the power of .
Step 5.3.135
Raise to the power of .
Step 5.3.136
Use the power rule to combine exponents.
Step 5.3.137
Add and .
Step 5.3.138
Multiply by .
Step 5.3.139
Multiply by .
Step 5.3.140
Combine and .
Step 5.3.141
Raise to the power of .
Step 5.3.142
Use the power rule to combine exponents.
Step 5.3.143
Add and .
Step 5.3.144
Add and .
Step 5.3.145
Add and .
Step 5.3.146
Reorder and .
Step 5.3.147
Reorder and .
Step 5.3.148
Move .
Step 5.3.149
Reorder and .
Step 6
Split the single integral into multiple integrals.
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Factor out .
Step 9
Using the Pythagorean Identity, rewrite as .
Step 10
Let . Then , so . Rewrite using and .
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Step 10.1
Let . Find .
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Step 10.1.1
Differentiate .
Step 10.1.2
The derivative of with respect to is .
Step 10.2
Rewrite the problem using and .
Step 11
Split the single integral into multiple integrals.
Step 12
Apply the constant rule.
Step 13
Since is constant with respect to , move out of the integral.
Step 14
By the Power Rule, the integral of with respect to is .
Step 15
Combine and .
Step 16
Since is constant with respect to , move out of the integral.
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
Since is constant with respect to , move out of the integral.
Step 29
Since is constant with respect to , move out of the integral.
Step 30
The integral of with respect to is .
Step 31
Simplify.
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Step 31.1
Simplify.
Step 31.2
Simplify.
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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 .
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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
Substitute back in for each integration substitution variable.
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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
Simplify.
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Step 33.1
Reduce the expression by cancelling the common factors.
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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 33.3
Combine and .
Step 34
Reorder terms.