Calculus Examples

Evaluate the Integral integral of 15(y^6+4y^3+3)^3(2y^5+4y^2) with respect to y
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Let . Then , so . Rewrite using and .
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Step 2.1
Let . Find .
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Step 2.1.1
Differentiate .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
Step 2.2
Rewrite the problem using and .
Step 3
Simplify.
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Step 3.1
Rewrite as .
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Step 3.1.1
Use to rewrite as .
Step 3.1.2
Apply the power rule and multiply exponents, .
Step 3.1.3
Combine and .
Step 3.1.4
Cancel the common factor of and .
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Step 3.1.4.1
Factor out of .
Step 3.1.4.2
Cancel the common factors.
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Step 3.1.4.2.1
Factor out of .
Step 3.1.4.2.2
Cancel the common factor.
Step 3.1.4.2.3
Rewrite the expression.
Step 3.1.4.2.4
Divide by .
Step 3.2
Rewrite as .
Step 3.3
Rewrite as .
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Step 3.3.1
Use to rewrite as .
Step 3.3.2
Apply the power rule and multiply exponents, .
Step 3.3.3
Combine and .
Step 3.3.4
Cancel the common factor of and .
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Step 3.3.4.1
Factor out of .
Step 3.3.4.2
Cancel the common factors.
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Step 3.3.4.2.1
Factor out of .
Step 3.3.4.2.2
Cancel the common factor.
Step 3.3.4.2.3
Rewrite the expression.
Step 3.3.4.2.4
Divide by .
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
Differentiate.
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Step 4.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3
Evaluate .
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Step 4.1.3.1
Use to rewrite as .
Step 4.1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.3
Differentiate using the Power Rule which states that is where .
Step 4.1.3.4
To write as a fraction with a common denominator, multiply by .
Step 4.1.3.5
Combine and .
Step 4.1.3.6
Combine the numerators over the common denominator.
Step 4.1.3.7
Simplify the numerator.
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Step 4.1.3.7.1
Multiply by .
Step 4.1.3.7.2
Subtract from .
Step 4.1.3.8
Combine and .
Step 4.1.3.9
Combine and .
Step 4.1.3.10
Multiply by .
Step 4.1.3.11
Factor out of .
Step 4.1.3.12
Cancel the common factors.
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Step 4.1.3.12.1
Factor out of .
Step 4.1.3.12.2
Cancel the common factor.
Step 4.1.3.12.3
Rewrite the expression.
Step 4.1.3.12.4
Divide by .
Step 4.1.4
Differentiate using the Constant Rule.
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Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Add and .
Step 4.1.5
Rewrite as a radical.
Step 4.2
Rewrite the problem using and .
Step 5
Combine and .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Simplify.
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Step 7.1
Combine and .
Step 7.2
Cancel the common factor of and .
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Step 7.2.1
Factor out of .
Step 7.2.2
Cancel the common factors.
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Step 7.2.2.1
Factor out of .
Step 7.2.2.2
Cancel the common factor.
Step 7.2.2.3
Rewrite the expression.
Step 7.2.2.4
Divide by .
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Simplify.
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Step 9.1
Rewrite as .
Step 9.2
Combine and .
Step 10
Substitute back in for each integration substitution variable.
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Step 10.1
Replace all occurrences of with .
Step 10.2
Replace all occurrences of with .