Calculus Examples

Evaluate the Integral integral from 1 to 2 of (( natural log of x)^2)/(x^3) with respect to x
Step 1
Apply basic rules of exponents.
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Step 1.1
Move out of the denominator by raising it to the power.
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
Multiply by .
Step 3.3
Raise to the power of .
Step 3.4
Use the power rule to combine exponents.
Step 3.5
Add and .
Step 4
Rewrite as .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Simplify the expression.
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Step 6.1
Simplify.
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Step 6.1.1
Cancel the common factor of .
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Step 6.1.1.1
Cancel the common factor.
Step 6.1.1.2
Rewrite the expression.
Step 6.1.2
Multiply by .
Step 6.1.3
Multiply by .
Step 6.2
Apply basic rules of exponents.
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Step 6.2.1
Move out of the denominator by raising it to the power.
Step 6.2.2
Multiply the exponents in .
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Step 6.2.2.1
Apply the power rule and multiply exponents, .
Step 6.2.2.2
Multiply by .
Step 7
Integrate by parts using the formula , where and .
Step 8
Simplify.
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Step 8.1
Combine and .
Step 8.2
Multiply by .
Step 8.3
Raise to the power of .
Step 8.4
Use the power rule to combine exponents.
Step 8.5
Add and .
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
Since is constant with respect to , move out of the integral.
Step 12
Apply basic rules of exponents.
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Step 12.1
Move out of the denominator by raising it to the power.
Step 12.2
Multiply the exponents in .
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Step 12.2.1
Apply the power rule and multiply exponents, .
Step 12.2.2
Multiply by .
Step 13
By the Power Rule, the integral of with respect to is .
Step 14
Simplify the answer.
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Step 14.1
Combine and .
Step 14.2
Substitute and simplify.
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Step 14.2.1
Evaluate at and at .
Step 14.2.2
Evaluate at and at .
Step 14.2.3
Simplify.
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Step 14.2.3.1
Raise to the power of .
Step 14.2.3.2
Multiply by .
Step 14.2.3.3
Raise to the power of .
Step 14.2.3.4
Multiply by .
Step 14.2.3.5
One to any power is one.
Step 14.2.3.6
Multiply by .
Step 14.2.3.7
One to any power is one.
Step 14.2.3.8
Multiply by .
Step 14.2.3.9
To write as a fraction with a common denominator, multiply by .
Step 14.2.3.10
Combine and .
Step 14.2.3.11
Combine the numerators over the common denominator.
Step 14.2.3.12
Multiply by .
Step 14.2.3.13
Rewrite the expression using the negative exponent rule .
Step 14.2.3.14
Raise to the power of .
Step 14.2.3.15
Multiply by .
Step 14.2.3.16
Multiply by .
Step 14.2.3.17
One to any power is one.
Step 14.2.3.18
Multiply by .
Step 14.2.3.19
To write as a fraction with a common denominator, multiply by .
Step 14.2.3.20
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 14.2.3.20.1
Multiply by .
Step 14.2.3.20.2
Multiply by .
Step 14.2.3.21
Combine the numerators over the common denominator.
Step 14.2.3.22
Add and .
Step 14.2.3.23
Multiply by .
Step 14.2.3.24
Multiply by .
Step 15
Simplify.
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Step 15.1
Combine the numerators over the common denominator.
Step 15.2
Simplify each term.
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Step 15.2.1
Combine the numerators over the common denominator.
Step 15.2.2
Simplify each term.
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Step 15.2.2.1
The natural logarithm of is .
Step 15.2.2.2
Raising to any positive power yields .
Step 15.2.2.3
Multiply by .
Step 15.2.2.4
The natural logarithm of is .
Step 15.2.2.5
Multiply by .
Step 15.2.3
Add and .
Step 15.2.4
Cancel the common factor of .
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Step 15.2.4.1
Factor out of .
Step 15.2.4.2
Cancel the common factor.
Step 15.2.4.3
Rewrite the expression.
Step 15.2.5
Multiply by .
Step 15.3
Add and .
Step 15.4
Simplify each term.
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Step 15.4.1
Factor out of .
Step 15.4.2
Move the negative in front of the fraction.
Step 15.5
To write as a fraction with a common denominator, multiply by .
Step 15.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 15.6.1
Multiply by .
Step 15.6.2
Multiply by .
Step 15.7
Combine the numerators over the common denominator.
Step 15.8
Simplify the numerator.
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Step 15.8.1
Apply the distributive property.
Step 15.8.2
Apply the distributive property.
Step 15.8.3
Multiply by .
Step 15.8.4
Multiply by .
Step 16
The result can be shown in multiple forms.
Exact Form:
Decimal Form: