Calculus Examples

Evaluate the Integral integral of x(4x-1)^4 with respect to x
Step 1
Simplify.
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Step 1.1
Use the Binomial Theorem.
Step 1.2
Simplify each term.
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Step 1.2.1
Apply the product rule to .
Step 1.2.2
Raise to the power of .
Step 1.2.3
Apply the product rule to .
Step 1.2.4
Multiply by by adding the exponents.
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Step 1.2.4.1
Move .
Step 1.2.4.2
Multiply by .
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Step 1.2.4.2.1
Raise to the power of .
Step 1.2.4.2.2
Use the power rule to combine exponents.
Step 1.2.4.3
Add and .
Step 1.2.5
Raise to the power of .
Step 1.2.6
Multiply by .
Step 1.2.7
Apply the product rule to .
Step 1.2.8
Raise to the power of .
Step 1.2.9
Multiply by .
Step 1.2.10
Raise to the power of .
Step 1.2.11
Multiply by .
Step 1.2.12
Multiply by .
Step 1.2.13
Raise to the power of .
Step 1.2.14
Multiply by .
Step 1.2.15
Raise to the power of .
Step 1.3
Apply the distributive property.
Step 1.4
Simplify.
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Step 1.4.1
Rewrite using the commutative property of multiplication.
Step 1.4.2
Rewrite using the commutative property of multiplication.
Step 1.4.3
Rewrite using the commutative property of multiplication.
Step 1.4.4
Rewrite using the commutative property of multiplication.
Step 1.4.5
Multiply by .
Step 1.5
Simplify each term.
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Step 1.5.1
Multiply by by adding the exponents.
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Step 1.5.1.1
Move .
Step 1.5.1.2
Multiply by .
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Step 1.5.1.2.1
Raise to the power of .
Step 1.5.1.2.2
Use the power rule to combine exponents.
Step 1.5.1.3
Add and .
Step 1.5.2
Multiply by by adding the exponents.
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Step 1.5.2.1
Move .
Step 1.5.2.2
Multiply by .
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Step 1.5.2.2.1
Raise to the power of .
Step 1.5.2.2.2
Use the power rule to combine exponents.
Step 1.5.2.3
Add and .
Step 1.5.3
Multiply by by adding the exponents.
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Step 1.5.3.1
Move .
Step 1.5.3.2
Multiply by .
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Step 1.5.3.2.1
Raise to the power of .
Step 1.5.3.2.2
Use the power rule to combine exponents.
Step 1.5.3.3
Add and .
Step 2
Simplify.
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Step 2.1
Raise to the power of .
Step 2.2
Raise to the power of .
Step 2.3
Use the power rule to combine exponents.
Step 2.4
Add and .
Step 3
Split the single integral into multiple integrals.
Step 4
Since is constant with respect to , move out of the integral.
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
By the Power Rule, the integral of with respect to is .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
By the Power Rule, the integral of with respect to is .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
By the Power Rule, the integral of with respect to is .
Step 13
Simplify.
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Step 13.1
Simplify.
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Step 13.1.1
Combine and .
Step 13.1.2
Combine and .
Step 13.1.3
Combine and .
Step 13.1.4
Combine and .
Step 13.2
Simplify.
Step 14
Reorder terms.