Calculus Examples

Evaluate the Integral integral of x(2x+5)^8 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
Raise to the power of .
Step 1.2.5
Multiply by .
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
Apply the product rule to .
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 1.2.16
Multiply by .
Step 1.2.17
Apply the product rule to .
Step 1.2.18
Raise to the power of .
Step 1.2.19
Multiply by .
Step 1.2.20
Raise to the power of .
Step 1.2.21
Multiply by .
Step 1.2.22
Apply the product rule to .
Step 1.2.23
Raise to the power of .
Step 1.2.24
Multiply by .
Step 1.2.25
Raise to the power of .
Step 1.2.26
Multiply by .
Step 1.2.27
Apply the product rule to .
Step 1.2.28
Raise to the power of .
Step 1.2.29
Multiply by .
Step 1.2.30
Raise to the power of .
Step 1.2.31
Multiply by .
Step 1.2.32
Multiply by .
Step 1.2.33
Raise to the power of .
Step 1.2.34
Multiply by .
Step 1.2.35
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
Rewrite using the commutative property of multiplication.
Step 1.4.6
Rewrite using the commutative property of multiplication.
Step 1.4.7
Rewrite using the commutative property of multiplication.
Step 1.4.8
Rewrite using the commutative property of multiplication.
Step 1.4.9
Move to the left of .
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 1.5.4
Multiply by by adding the exponents.
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Step 1.5.4.1
Move .
Step 1.5.4.2
Multiply by .
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Step 1.5.4.2.1
Raise to the power of .
Step 1.5.4.2.2
Use the power rule to combine exponents.
Step 1.5.4.3
Add and .
Step 1.5.5
Multiply by by adding the exponents.
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Step 1.5.5.1
Move .
Step 1.5.5.2
Multiply by .
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Step 1.5.5.2.1
Raise to the power of .
Step 1.5.5.2.2
Use the power rule to combine exponents.
Step 1.5.5.3
Add and .
Step 1.5.6
Multiply by by adding the exponents.
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Step 1.5.6.1
Move .
Step 1.5.6.2
Multiply by .
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Step 1.5.6.2.1
Raise to the power of .
Step 1.5.6.2.2
Use the power rule to combine exponents.
Step 1.5.6.3
Add and .
Step 1.5.7
Multiply by by adding the exponents.
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Step 1.5.7.1
Move .
Step 1.5.7.2
Multiply by .
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Step 1.5.7.2.1
Raise to the power of .
Step 1.5.7.2.2
Use the power rule to combine exponents.
Step 1.5.7.3
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
Since is constant with respect to , move out of the integral.
Step 13
By the Power Rule, the integral of with respect to is .
Step 14
Since is constant with respect to , move out of the integral.
Step 15
By the Power Rule, the integral of with respect to is .
Step 16
Since is constant with respect to , move out of the integral.
Step 17
By the Power Rule, the integral of with respect to is .
Step 18
Since is constant with respect to , move out of the integral.
Step 19
By the Power Rule, the integral of with respect to is .
Step 20
Since is constant with respect to , move out of the integral.
Step 21
By the Power Rule, the integral of with respect to is .
Step 22
Simplify.
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Step 22.1
Simplify.
Step 22.2
Simplify.
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Step 22.2.1
Combine and .
Step 22.2.2
Combine and .
Step 22.3
Reorder terms.