Calculus Examples

Integrate Using u-Substitution
,
Step 1
Let . Then . 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
By the Sum Rule, the derivative of with respect to is .
Step 1.1.3
Evaluate .
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Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Differentiate using the Constant Rule.
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Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Add and .
Step 1.2
Rewrite the problem using and .
Step 2
Since is constant with respect to , move out of the integral.
Step 3
Apply basic rules of exponents.
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Step 3.1
Move out of the denominator by raising it to the power.
Step 3.2
Multiply the exponents in .
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Step 3.2.1
Apply the power rule and multiply exponents, .
Step 3.2.2
Multiply by .
Step 4
By the Power Rule, the integral of with respect to is .
Step 5
Simplify.
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Step 5.1
Simplify.
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Step 5.1.1
Combine and .
Step 5.1.2
Move to the denominator using the negative exponent rule .
Step 5.2
Simplify.
Step 5.3
Simplify.
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Step 5.3.1
Multiply by .
Step 5.3.2
Combine and .
Step 5.3.3
Move the negative in front of the fraction.
Step 6
Replace all occurrences of with .
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