Calculus Examples

Evaluate the Integral integral of x^2 square root of x+1 with respect to x
Step 1
Integrate by parts using the formula , where and .
Step 2
Simplify.
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Step 2.1
Combine and .
Step 2.2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Simplify.
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Step 4.1
Combine and .
Step 4.2
Multiply by .
Step 5
Let . Then . Rewrite using and .
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Step 5.1
Let . Find .
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Step 5.1.1
Differentiate .
Step 5.1.2
By the Sum Rule, the derivative of with respect to is .
Step 5.1.3
Differentiate using the Power Rule which states that is where .
Step 5.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.5
Add and .
Step 5.2
Rewrite the problem using and .
Step 6
Expand .
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Step 6.1
Apply the distributive property.
Step 6.2
Reorder and .
Step 6.3
Raise to the power of .
Step 6.4
Use the power rule to combine exponents.
Step 6.5
Write as a fraction with a common denominator.
Step 6.6
Combine the numerators over the common denominator.
Step 6.7
Add and .
Step 7
Rewrite as .
Step 8
Split the single integral into multiple integrals.
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
Simplify.
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Step 12.1
Simplify.
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Step 12.1.1
Combine and .
Step 12.1.2
Combine and .
Step 12.2
Simplify.
Step 12.3
Simplify.
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Step 12.3.1
Combine and .
Step 12.3.2
Combine and .
Step 12.3.3
To write as a fraction with a common denominator, multiply by .
Step 12.3.4
Combine and .
Step 12.3.5
Combine the numerators over the common denominator.
Step 12.3.6
Combine and .
Step 12.3.7
Combine and .
Step 12.3.8
Multiply by .
Step 12.3.9
Combine and .
Step 12.3.10
Multiply by .
Step 12.3.11
Cancel the common factor of and .
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Step 12.3.11.1
Factor out of .
Step 12.3.11.2
Cancel the common factors.
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Step 12.3.11.2.1
Factor out of .
Step 12.3.11.2.2
Cancel the common factor.
Step 12.3.11.2.3
Rewrite the expression.
Step 12.3.11.2.4
Divide by .
Step 12.3.12
Move to the left of .
Step 13
Replace all occurrences of with .
Step 14
Reorder terms.