Calculus Examples

Evaluate the Integral integral from 0 to 5 of 1/( square root of 3x+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
Add and .
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
Add and .
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
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
Substitute and simplify.
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Step 6.1
Evaluate at and at .
Step 6.2
Simplify.
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Step 6.2.1
Rewrite as .
Step 6.2.2
Apply the power rule and multiply exponents, .
Step 6.2.3
Cancel the common factor of .
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Step 6.2.3.1
Cancel the common factor.
Step 6.2.3.2
Rewrite the expression.
Step 6.2.4
Evaluate the exponent.
Step 6.2.5
Multiply by .
Step 6.2.6
One to any power is one.
Step 6.2.7
Multiply by .
Step 6.2.8
Subtract from .
Step 6.2.9
Combine and .
Step 6.2.10
Cancel the common factor of and .
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Step 6.2.10.1
Factor out of .
Step 6.2.10.2
Cancel the common factors.
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Step 6.2.10.2.1
Factor out of .
Step 6.2.10.2.2
Cancel the common factor.
Step 6.2.10.2.3
Rewrite the expression.
Step 6.2.10.2.4
Divide by .
Step 7