Question

section 3.6: additional integr
score: 40/60 answered: 4/6
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question 5

compute the given integral.

$$ int x ^ { 4 } ln x d x = $$

Explanation:

Step1: Use integration by parts formula

The integration by parts formula is \(\int u\mathrm{d}v=uv - \int v\mathrm{d}u\). Let \(u = \ln x\), then \(\mathrm{d}u=\frac{1}{x}\mathrm{d}x\). Let \(\mathrm{d}v=x^{4}\mathrm{d}x\), then \(v=\frac{x^{5}}{5}\).

Step2: Apply the formula

$$
LATEXBLOCK0
$$

Step3: Integrate \(x^{4}\)

We know that \(\int x^{n}\mathrm{d}x=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). So \(\int x^{4}\mathrm{d}x=\frac{x^{5}}{5}+C\).

$$
LATEXBLOCK1
$$

Answer:

\(\frac{x^{5}}{5}\ln x-\frac{x^{5}}{25}+C\)