Error Analysis In Integration By Parts

Question

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?

Accepted Answer

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.

Answer

The error lies in neglecting the constant of integration, which does not significantly affect the final answer.

Answer

There is no mistake since any proper integration by parts yields the same result.

Answer

The error is in the choice of $$u$$ and $$dv$$; the better choice would be $$u=\ln(x)$$ and $$dv=x\,dx$$.

Error Analysis In Integration By Parts

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