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Calculus Examples
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-∫vdu∫udv=uv−∫vdu, where u=ln(x)u=ln(x) and dv=1dv=1.
ln(x)x-∫x1xdxln(x)x−∫x1xdx
Step 5
Step 5.1
Combine xx and 1x1x.
ln(x)x-∫xxdxln(x)x−∫xxdx
Step 5.2
Cancel the common factor of xx.
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