Calculus Examples

Evaluate the Integral integral of 1/( square root of 2x) with respect to x
Step 1
Let . Then , so . 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Multiply by .
Step 1.2
Rewrite the problem using and .
Step 2
Simplify.
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Step 2.1
Multiply by .
Step 2.2
Move to the left of .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Apply basic rules of exponents.
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Step 4.1
Use to rewrite as .
Step 4.2
Move out of the denominator by raising it to the power.
Step 4.3
Multiply the exponents in .
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Step 4.3.1
Apply the power rule and multiply exponents, .
Step 4.3.2
Combine and .
Step 4.3.3
Move the negative in front of the fraction.
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Simplify.
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Step 6.1
Rewrite as .
Step 6.2
Simplify.
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Step 6.2.1
Combine and .
Step 6.2.2
Cancel the common factor of .
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Step 6.2.2.1
Cancel the common factor.
Step 6.2.2.2
Rewrite the expression.
Step 6.2.3
Multiply by .
Step 7
Replace all occurrences of with .