APFIVE Error Analysis In Integration By Parts
A researcher attempts to compute $$\int x\ln(x)\,dx$$ by choosing $$u = x$$ and $$dv = \ln(x)\,dx$$. They then determine that $$du = dx$$ and incorrectly assert that $$\int \ln(x)\,dx = x\ln(x)$$. What is the error in this process?
A
The error is in the incorrect integration of $$\ln(x)\,dx$$; the proper antiderivative is $$x\ln(x) - x + C$$, and this mistake causes an incorrect final answer.
B
There is no mistake since any proper integration by parts yields the same result.
C
The error lies in neglecting the constant of integration, which does not significantly affect the final answer.
D
The error is in the choice of $$u$$ and $$dv$$; the better choice would be $$u=\ln(x)$$ and $$dv=x\,dx$$.
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