APFIVE To evaluate $$\int x\ln(x)\,dx$$ using integration by parts, a student sets $$u = x$$ and $$dv = \ln(x)\,dx$$. The student finds that $$du = dx$$ and incorrectly determines that $$v = \int \ln(x)\,dx = x\ln(x)$$. Which of the following describes the error in this procedure?
The error is in the choice of $$u$$ and $$dv$$; the better choice would be $$u=\ln(x)$$ and $$dv=x\,dx$$.
The error lies in neglecting the constant of integration, which does not significantly affect the final answer.
There is no mistake since any proper integration by parts yields the same result.
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.
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