Calculus Examples

Evaluate the Integral integral of x/( square root of 2x-1) 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
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
Simplify.
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Step 2.1
Multiply by .
Step 2.2
Combine.
Step 2.3
Apply the distributive property.
Step 2.4
Cancel the common factor of .
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Step 2.4.1
Cancel the common factor.
Step 2.4.2
Rewrite the expression.
Step 2.5
Cancel the common factor of .
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Step 2.5.1
Cancel the common factor.
Step 2.5.2
Rewrite the expression.
Step 2.6
Multiply by .
Step 2.7
Multiply by .
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
Expand .
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Step 5.1
Apply the distributive property.
Step 5.2
Raise to the power of .
Step 5.3
Use the power rule to combine exponents.
Step 5.4
Write as a fraction with a common denominator.
Step 5.5
Combine the numerators over the common denominator.
Step 5.6
Subtract from .
Step 5.7
Multiply by .
Step 6
Split the single integral into multiple integrals.
Step 7
By the Power Rule, the integral of with respect to is .
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Simplify.
Step 10
Replace all occurrences of with .
Step 11
Simplify.
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Step 11.1
Combine and .
Step 11.2
To write as a fraction with a common denominator, multiply by .
Step 11.3
Combine and .
Step 11.4
Combine the numerators over the common denominator.
Step 11.5
Simplify the numerator.
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Step 11.5.1
Factor out of .
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Step 11.5.1.1
Move .
Step 11.5.1.2
Factor out of .
Step 11.5.1.3
Factor out of .
Step 11.5.1.4
Factor out of .
Step 11.5.2
Divide by .
Step 11.5.3
Simplify.
Step 11.5.4
Add and .
Step 11.5.5
Factor out of .
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Step 11.5.5.1
Factor out of .
Step 11.5.5.2
Factor out of .
Step 11.5.5.3
Factor out of .
Step 11.5.6
Multiply by .
Step 11.6
Combine.
Step 11.7
Cancel the common factor.
Step 11.8
Rewrite the expression.
Step 11.9
Multiply by .