Calculus Examples

Find the Antiderivative natural log of x
ln(x)ln(x)
Step 1
Write ln(x)ln(x) as a function.
f(x)=ln(x)f(x)=ln(x)
Step 2
The function F(x)F(x) can be found by finding the indefinite integral of the derivative f(x)f(x).
F(x)=f(x)dxF(x)=f(x)dx
Step 3
Set up the integral to solve.
F(x)=ln(x)dxF(x)=ln(x)dx
Step 4
Integrate by parts using the formula udv=uv-vduudv=uvvdu, where u=ln(x)u=ln(x) and dv=1dv=1.
ln(x)x-x1xdxln(x)xx1xdx
Step 5
Simplify.
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Step 5.1
Combine xx and 1x1x.
ln(x)x-xxdxln(x)xxxdx
Step 5.2
Cancel the common factor of xx.
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Step 5.2.1
Cancel the common factor.
ln(x)x-xxdx
Step 5.2.2
Rewrite the expression.
ln(x)x-dx
ln(x)x-dx
ln(x)x-dx
Step 6
Apply the constant rule.
ln(x)x-(x+C)
Step 7
Simplify.
ln(x)x-x+C
Step 8
The answer is the antiderivative of the function f(x)=ln(x).
F(x)=ln(x)x-x+C
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