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The derivative is f′(x)/f(x)f'(x)/f(x)f′(x)/f(x).
You use the chain rule for differentiation.
It is the derivative of ln∣f(x)∣\ln|f(x)|ln∣f(x)∣.
The integral is ln∣f(x)∣+C\ln|f(x)| + Cln∣f(x)∣+C.
f′(x)=2xf'(x) = 2xf′(x)=2x.
ln(x2+1)+C\ln(x^2 + 1) + Cln(x2+1)+C; u=x2+1u = x^2 + 1u=x2+1 for substitution.
It converts products into sums to simplify integration.
It's f′(x)/f(x)f'(x)/f(x)f′(x)/f(x) by the definition of logarithmic derivative.
Look for f′(x)/f(x)f'(x)/f(x)f′(x)/f(x) in expressions.
CCC is the constant of integration.
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