Calculus Examples

Evaluate the Integral integral from 0 to 1 of y/(e^(2y)) with respect to y
Step 1
Simplify the expression.
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Step 1.1
Negate the exponent of and move it out of the denominator.
Step 1.2
Multiply the exponents in .
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Step 1.2.1
Apply the power rule and multiply exponents, .
Step 1.2.2
Multiply by .
Step 2
Integrate by parts using the formula , where and .
Step 3
Simplify.
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Step 3.1
Combine and .
Step 3.2
Combine and .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Simplify.
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Step 5.1
Multiply by .
Step 5.2
Multiply by .
Step 6
Let . Then , so . Rewrite using and .
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Step 6.1
Let . Find .
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Step 6.1.1
Differentiate .
Step 6.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.3
Differentiate using the Power Rule which states that is where .
Step 6.1.4
Multiply by .
Step 6.2
Substitute the lower limit in for in .
Step 6.3
Multiply by .
Step 6.4
Substitute the upper limit in for in .
Step 6.5
Multiply by .
Step 6.6
The values found for and will be used to evaluate the definite integral.
Step 6.7
Rewrite the problem using , , and the new limits of integration.
Step 7
Simplify.
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Step 7.1
Move the negative in front of the fraction.
Step 7.2
Combine and .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Simplify.
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Step 10.1
Multiply by .
Step 10.2
Multiply by .
Step 11
The integral of with respect to is .
Step 12
Substitute and simplify.
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Step 12.1
Evaluate at and at .
Step 12.2
Evaluate at and at .
Step 12.3
Simplify.
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Step 12.3.1
Multiply by .
Step 12.3.2
Multiply by .
Step 12.3.3
Move to the denominator using the negative exponent rule .
Step 12.3.4
Multiply by .
Step 12.3.5
Anything raised to is .
Step 12.3.6
Multiply by .
Step 12.3.7
Cancel the common factor of and .
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Step 12.3.7.1
Factor out of .
Step 12.3.7.2
Cancel the common factors.
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Step 12.3.7.2.1
Factor out of .
Step 12.3.7.2.2
Cancel the common factor.
Step 12.3.7.2.3
Rewrite the expression.
Step 12.3.7.2.4
Divide by .
Step 12.3.8
Add and .
Step 12.3.9
Anything raised to is .
Step 12.3.10
Multiply by .
Step 12.3.11
To write as a fraction with a common denominator, multiply by .
Step 12.3.12
Combine and .
Step 12.3.13
Combine the numerators over the common denominator.
Step 12.3.14
Multiply by .
Step 12.3.15
Combine and .
Step 12.3.16
Combine and .
Step 12.3.17
Move to the left of .
Step 12.3.18
Cancel the common factor of and .
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Step 12.3.18.1
Factor out of .
Step 12.3.18.2
Cancel the common factors.
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Step 12.3.18.2.1
Factor out of .
Step 12.3.18.2.2
Cancel the common factor.
Step 12.3.18.2.3
Rewrite the expression.
Step 12.3.19
Move the negative in front of the fraction.
Step 13
Simplify.
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Step 13.1
Rewrite as .
Step 13.2
Factor out of .
Step 13.3
Factor out of .
Step 13.4
Move the negative in front of the fraction.
Step 14
Simplify.
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Step 14.1
Apply the distributive property.
Step 14.2
Multiply .
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Step 14.2.1
Combine and .
Step 14.2.2
Multiply by by adding the exponents.
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Step 14.2.2.1
Use the power rule to combine exponents.
Step 14.2.2.2
Subtract from .
Step 14.2.3
Simplify .
Step 14.3
Combine and .
Step 14.4
Move to the left of .
Step 14.5
Rewrite as .
Step 14.6
Simplify the numerator.
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Step 14.6.1
Write as a fraction with a common denominator.
Step 14.6.2
Combine the numerators over the common denominator.
Step 14.6.3
Add and .
Step 14.6.4
Combine the numerators over the common denominator.
Step 14.7
Multiply the numerator by the reciprocal of the denominator.
Step 14.8
Multiply .
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Step 14.8.1
Multiply by .
Step 14.8.2
Multiply by .
Step 15
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 16