Calculus Examples

Evaluate the Integral integral from 1 to 5 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
Substitute the lower limit in for in .
Step 1.3
Simplify.
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Step 1.3.1
Multiply by .
Step 1.3.2
Subtract from .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Simplify.
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Step 1.5.1
Multiply by .
Step 1.5.2
Subtract from .
Step 1.6
The values found for and will be used to evaluate the definite integral.
Step 1.7
Rewrite the problem using , , and the new limits of integration.
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
Combine and .
Step 10
Substitute and simplify.
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Step 10.1
Evaluate at and at .
Step 10.2
Simplify.
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Step 10.2.1
Rewrite as .
Step 10.2.2
Apply the power rule and multiply exponents, .
Step 10.2.3
Cancel the common factor of .
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Step 10.2.3.1
Cancel the common factor.
Step 10.2.3.2
Rewrite the expression.
Step 10.2.4
Raise to the power of .
Step 10.2.5
Combine and .
Step 10.2.6
Multiply by .
Step 10.2.7
Cancel the common factor of and .
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Step 10.2.7.1
Factor out of .
Step 10.2.7.2
Cancel the common factors.
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Step 10.2.7.2.1
Factor out of .
Step 10.2.7.2.2
Cancel the common factor.
Step 10.2.7.2.3
Rewrite the expression.
Step 10.2.7.2.4
Divide by .
Step 10.2.8
Rewrite as .
Step 10.2.9
Apply the power rule and multiply exponents, .
Step 10.2.10
Cancel the common factor of .
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Step 10.2.10.1
Cancel the common factor.
Step 10.2.10.2
Rewrite the expression.
Step 10.2.11
Evaluate the exponent.
Step 10.2.12
Multiply by .
Step 10.2.13
Add and .
Step 10.2.14
One to any power is one.
Step 10.2.15
Multiply by .
Step 10.2.16
One to any power is one.
Step 10.2.17
Multiply by .
Step 10.2.18
To write as a fraction with a common denominator, multiply by .
Step 10.2.19
Combine and .
Step 10.2.20
Combine the numerators over the common denominator.
Step 10.2.21
Simplify the numerator.
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Step 10.2.21.1
Multiply by .
Step 10.2.21.2
Add and .
Step 10.2.22
To write as a fraction with a common denominator, multiply by .
Step 10.2.23
Combine and .
Step 10.2.24
Combine the numerators over the common denominator.
Step 10.2.25
Simplify the numerator.
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Step 10.2.25.1
Multiply by .
Step 10.2.25.2
Subtract from .
Step 10.2.26
Multiply by .
Step 10.2.27
Multiply by .
Step 10.2.28
Cancel the common factor of and .
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Step 10.2.28.1
Factor out of .
Step 10.2.28.2
Cancel the common factors.
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Step 10.2.28.2.1
Factor out of .
Step 10.2.28.2.2
Cancel the common factor.
Step 10.2.28.2.3
Rewrite the expression.
Step 11
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 12