Calculus Examples

Evaluate the Integral integral from 0 to pi/18 of sin(9x)^5 with respect to x
Step 1
Let . Then , so . Rewrite using and .
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Step 1.1
Let . Find .
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Step 1.1.1
Differentiate .
Step 1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Multiply by .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Multiply by .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Cancel the common factor of .
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Step 1.5.1
Factor out of .
Step 1.5.2
Cancel the common factor.
Step 1.5.3
Rewrite the expression.
Step 1.6
The values found for and will be used to evaluate the definite integral.
Step 1.7
Rewrite the problem using , , and the new limits of integration.
Step 2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Factor out .
Step 5
Simplify with factoring out.
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Step 5.1
Factor out of .
Step 5.2
Rewrite as exponentiation.
Step 6
Using the Pythagorean Identity, rewrite as .
Step 7
Let . Then , so . Rewrite using and .
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Step 7.1
Let . Find .
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Step 7.1.1
Differentiate .
Step 7.1.2
The derivative of with respect to is .
Step 7.2
Substitute the lower limit in for in .
Step 7.3
The exact value of is .
Step 7.4
Substitute the upper limit in for in .
Step 7.5
The exact value of is .
Step 7.6
The values found for and will be used to evaluate the definite integral.
Step 7.7
Rewrite the problem using , , and the new limits of integration.
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Expand .
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Step 9.1
Rewrite as .
Step 9.2
Apply the distributive property.
Step 9.3
Apply the distributive property.
Step 9.4
Apply the distributive property.
Step 9.5
Move .
Step 9.6
Move .
Step 9.7
Multiply by .
Step 9.8
Multiply by .
Step 9.9
Multiply by .
Step 9.10
Multiply by .
Step 9.11
Multiply by .
Step 9.12
Use the power rule to combine exponents.
Step 9.13
Add and .
Step 9.14
Subtract from .
Step 9.15
Reorder and .
Step 9.16
Move .
Step 10
Split the single integral into multiple integrals.
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
Combine and .
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
Apply the constant rule.
Step 17
Combine and .
Step 18
Substitute and simplify.
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Step 18.1
Evaluate at and at .
Step 18.2
Evaluate at and at .
Step 18.3
Simplify.
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Step 18.3.1
Raising to any positive power yields .
Step 18.3.2
Cancel the common factor of and .
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Step 18.3.2.1
Factor out of .
Step 18.3.2.2
Cancel the common factors.
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Step 18.3.2.2.1
Factor out of .
Step 18.3.2.2.2
Cancel the common factor.
Step 18.3.2.2.3
Rewrite the expression.
Step 18.3.2.2.4
Divide by .
Step 18.3.3
Add and .
Step 18.3.4
One to any power is one.
Step 18.3.5
Write as a fraction with a common denominator.
Step 18.3.6
Combine the numerators over the common denominator.
Step 18.3.7
Add and .
Step 18.3.8
Subtract from .
Step 18.3.9
Raising to any positive power yields .
Step 18.3.10
Cancel the common factor of and .
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Step 18.3.10.1
Factor out of .
Step 18.3.10.2
Cancel the common factors.
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Step 18.3.10.2.1
Factor out of .
Step 18.3.10.2.2
Cancel the common factor.
Step 18.3.10.2.3
Rewrite the expression.
Step 18.3.10.2.4
Divide by .
Step 18.3.11
One to any power is one.
Step 18.3.12
Subtract from .
Step 18.3.13
Multiply by .
Step 18.3.14
Combine and .
Step 18.3.15
To write as a fraction with a common denominator, multiply by .
Step 18.3.16
To write as a fraction with a common denominator, multiply by .
Step 18.3.17
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 18.3.17.1
Multiply by .
Step 18.3.17.2
Multiply by .
Step 18.3.17.3
Multiply by .
Step 18.3.17.4
Multiply by .
Step 18.3.18
Combine the numerators over the common denominator.
Step 18.3.19
Simplify the numerator.
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Step 18.3.19.1
Multiply by .
Step 18.3.19.2
Multiply by .
Step 18.3.19.3
Add and .
Step 18.3.20
Move the negative in front of the fraction.
Step 18.3.21
Multiply by .
Step 18.3.22
Multiply by .
Step 18.3.23
Multiply by .
Step 18.3.24
Multiply by .
Step 19
The result can be shown in multiple forms.
Exact Form:
Decimal Form: