Calculus Examples

Evaluate the Integral integral from 1 to 7 of (( natural log of x)^2)/(x^3) with respect to x
Step 1
Apply basic rules of exponents.
Step 1.1
Move out of the denominator by raising it to the power.
Step 1.2
Multiply the exponents in .
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.
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.
Step 6.1
Simplify.
Step 6.1.1
Cancel the common factor of .
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.
Step 6.2.1
Move out of the denominator by raising it to the power.
Step 6.2.2
Multiply the exponents in .
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.
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.
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.
Step 12.1
Move out of the denominator by raising it to the power.
Step 12.2
Multiply the exponents in .
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.
Step 14.1
Combine and .
Step 14.2
Substitute and simplify.
Step 14.2.1
Evaluate at and at .
Step 14.2.2
Evaluate at and at .
Step 14.2.3
Simplify.
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 .
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
Cancel the common factor of and .
Step 14.2.3.23.1
Factor out of .
Step 14.2.3.23.2
Cancel the common factors.
Step 14.2.3.23.2.1
Factor out of .
Step 14.2.3.23.2.2
Cancel the common factor.
Step 14.2.3.23.2.3
Rewrite the expression.
Step 14.2.3.24
Multiply by .
Step 14.2.3.25
Multiply by .
Step 14.2.3.26
Cancel the common factor of and .
Step 14.2.3.26.1
Factor out of .
Step 14.2.3.26.2
Cancel the common factors.
Step 14.2.3.26.2.1
Factor out of .
Step 14.2.3.26.2.2
Cancel the common factor.
Step 14.2.3.26.2.3
Rewrite the expression.
Step 15
Simplify.
Step 15.1
Combine the numerators over the common denominator.
Step 15.2
Simplify each term.
Step 15.2.1
Combine the numerators over the common denominator.
Step 15.2.2
Simplify each term.
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 .
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.
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 .
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.
Step 15.8.1
Apply the distributive property.
Step 15.8.2
Multiply by .
Step 16
The result can be shown in multiple forms.
Exact Form:
Decimal Form: