Calculus Examples

Evaluate the Integral integral from 1 to 4 of (e^( square root of x))/( square root of x) with respect to x
Step 1
Apply basic rules of exponents.
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Step 1.1
Use to rewrite as .
Step 1.2
Use to rewrite as .
Step 1.3
Move out of the denominator by raising it to the power.
Step 1.4
Multiply the exponents in .
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Step 1.4.1
Apply the power rule and multiply exponents, .
Step 1.4.2
Combine and .
Step 1.4.3
Move the negative in front of the fraction.
Step 2
Let . Then , so . Rewrite using and .
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Step 2.1
Let . Find .
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Step 2.1.1
Differentiate .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3
To write as a fraction with a common denominator, multiply by .
Step 2.1.4
Combine and .
Step 2.1.5
Combine the numerators over the common denominator.
Step 2.1.6
Simplify the numerator.
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Step 2.1.6.1
Multiply by .
Step 2.1.6.2
Subtract from .
Step 2.1.7
Move the negative in front of the fraction.
Step 2.1.8
Simplify.
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Step 2.1.8.1
Rewrite the expression using the negative exponent rule .
Step 2.1.8.2
Multiply by .
Step 2.2
Substitute the lower limit in for in .
Step 2.3
One to any power is one.
Step 2.4
Substitute the upper limit in for in .
Step 2.5
Simplify.
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Step 2.5.1
Rewrite as .
Step 2.5.2
Apply the power rule and multiply exponents, .
Step 2.5.3
Cancel the common factor of .
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Step 2.5.3.1
Cancel the common factor.
Step 2.5.3.2
Rewrite the expression.
Step 2.5.4
Evaluate the exponent.
Step 2.6
The values found for and will be used to evaluate the definite integral.
Step 2.7
Rewrite the problem using , , and the new limits of integration.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
The integral of with respect to is .
Step 5
Substitute and simplify.
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Step 5.1
Evaluate at and at .
Step 5.2
Simplify.
Step 6
Simplify.
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Step 6.1
Apply the distributive property.
Step 6.2
Multiply by .
Step 7
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 8