Calculus Examples

Evaluate the Integral integral from 0 to pi/2 of sin(x)^4 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
Substitute the lower limit in for in .
Step 3.3
Multiply by .
Step 3.4
Substitute the upper limit in for in .
Step 3.5
Cancel the common factor of .
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Step 3.5.1
Cancel the common factor.
Step 3.5.2
Rewrite the expression.
Step 3.6
The values found for and will be used to evaluate the definite integral.
Step 3.7
Rewrite the problem using , , and the new limits of integration.
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
Apply the distributive property.
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
Reorder and .
Step 5.2.8
Reorder and .
Step 5.2.9
Move .
Step 5.2.10
Reorder and .
Step 5.2.11
Reorder and .
Step 5.2.12
Move parentheses.
Step 5.2.13
Move .
Step 5.2.14
Reorder and .
Step 5.2.15
Reorder and .
Step 5.2.16
Move .
Step 5.2.17
Move .
Step 5.2.18
Reorder and .
Step 5.2.19
Reorder and .
Step 5.2.20
Move parentheses.
Step 5.2.21
Move .
Step 5.2.22
Move .
Step 5.2.23
Multiply by .
Step 5.2.24
Multiply by .
Step 5.2.25
Multiply by .
Step 5.2.26
Multiply by .
Step 5.2.27
Multiply by .
Step 5.2.28
Combine and .
Step 5.2.29
Multiply by .
Step 5.2.30
Combine and .
Step 5.2.31
Multiply by .
Step 5.2.32
Combine and .
Step 5.2.33
Combine and .
Step 5.2.34
Multiply by .
Step 5.2.35
Multiply by .
Step 5.2.36
Multiply by .
Step 5.2.37
Combine and .
Step 5.2.38
Multiply by .
Step 5.2.39
Multiply by .
Step 5.2.40
Combine and .
Step 5.2.41
Raise to the power of .
Step 5.2.42
Raise to the power of .
Step 5.2.43
Use the power rule to combine exponents.
Step 5.2.44
Add and .
Step 5.2.45
Subtract from .
Step 5.2.46
Combine and .
Step 5.2.47
Reorder and .
Step 5.2.48
Reorder and .
Step 5.3
Simplify.
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Step 5.3.1
Cancel the common factor of and .
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Step 5.3.1.1
Factor out of .
Step 5.3.1.2
Cancel the common factors.
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Step 5.3.1.2.1
Factor out of .
Step 5.3.1.2.2
Cancel the common factor.
Step 5.3.1.2.3
Rewrite the expression.
Step 5.3.2
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
Use the half-angle formula to rewrite as .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Simplify.
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Step 10.1
Multiply by .
Step 10.2
Multiply by .
Step 11
Split the single integral into multiple integrals.
Step 12
Apply the constant rule.
Step 13
Let . Then , so . Rewrite using and .
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Step 13.1
Let . Find .
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Step 13.1.1
Differentiate .
Step 13.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 13.1.3
Differentiate using the Power Rule which states that is where .
Step 13.1.4
Multiply by .
Step 13.2
Substitute the lower limit in for in .
Step 13.3
Multiply by .
Step 13.4
Substitute the upper limit in for in .
Step 13.5
The values found for and will be used to evaluate the definite integral.
Step 13.6
Rewrite the problem using , , and the new limits of integration.
Step 14
Combine and .
Step 15
Since is constant with respect to , move out of the integral.
Step 16
The integral of with respect to is .
Step 17
Combine and .
Step 18
Apply the constant rule.
Step 19
Combine and .
Step 20
Since is constant with respect to , move out of the integral.
Step 21
Since is constant with respect to , move out of the integral.
Step 22
The integral of with respect to is .
Step 23
Simplify.
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Step 23.1
Combine and .
Step 23.2
To write as a fraction with a common denominator, multiply by .
Step 23.3
Combine and .
Step 23.4
Combine the numerators over the common denominator.
Step 23.5
Combine and .
Step 23.6
Cancel the common factor of and .
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Step 23.6.1
Factor out of .
Step 23.6.2
Cancel the common factors.
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Step 23.6.2.1
Factor out of .
Step 23.6.2.2
Cancel the common factor.
Step 23.6.2.3
Rewrite the expression.
Step 24
Substitute and simplify.
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Step 24.1
Evaluate at and at .
Step 24.2
Evaluate at and at .
Step 24.3
Evaluate at and at .
Step 24.4
Evaluate at and at .
Step 24.5
Simplify.
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Step 24.5.1
Add and .
Step 24.5.2
Cancel the common factor of and .
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Step 24.5.2.1
Factor out of .
Step 24.5.2.2
Cancel the common factors.
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Step 24.5.2.2.1
Factor out of .
Step 24.5.2.2.2
Cancel the common factor.
Step 24.5.2.2.3
Rewrite the expression.
Step 24.5.2.2.4
Divide by .
Step 24.5.3
Multiply by .
Step 24.5.4
Add and .
Step 25
Simplify.
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Step 25.1
The exact value of is .
Step 25.2
The exact value of is .
Step 25.3
Multiply by .
Step 25.4
Add and .
Step 25.5
Multiply by .
Step 25.6
Add and .
Step 26
Simplify.
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Step 26.1
Simplify each term.
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Step 26.1.1
Simplify the numerator.
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Step 26.1.1.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 26.1.1.2
The exact value of is .
Step 26.1.2
Divide by .
Step 26.2
Add and .
Step 26.3
Combine and .
Step 26.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 26.5
The exact value of is .
Step 26.6
Multiply by .
Step 26.7
Add and .
Step 26.8
Simplify each term.
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Step 26.8.1
Multiply the numerator by the reciprocal of the denominator.
Step 26.8.2
Multiply .
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Step 26.8.2.1
Multiply by .
Step 26.8.2.2
Multiply by .
Step 26.9
To write as a fraction with a common denominator, multiply by .
Step 26.10
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 26.10.1
Multiply by .
Step 26.10.2
Multiply by .
Step 26.11
Combine the numerators over the common denominator.
Step 26.12
Move to the left of .
Step 26.13
Add and .
Step 26.14
Multiply .
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Step 26.14.1
Multiply by .
Step 26.14.2
Multiply by .
Step 27
The result can be shown in multiple forms.
Exact Form:
Decimal Form: