Question

use the substitution formula, \\(\int_{a}^{b} f(g(x)) \cdot g(x) \\, dx = \int_{g(a)}^{g(b)} f(u) \\, du\\) where \\(g(x) = u\\), to evaluate the following integral. \\(\int_{1}^{2} \frac{3(\ln x)^2}{x} \\, dx\\) \\(\int_{1}^{2} \frac{3(\ln x)^2}{x} \\, dx = \square\\) (type an exact answer.)

Explanation:

Step1: Choose substitution

Let \( u = \ln x \), then \( du=\frac{1}{x}dx \).
When \( x = 1 \), \( u=\ln 1 = 0 \); when \( x = 2 \), \( u=\ln 2 \).

Step2: Rewrite integral

The integral \( \int_{1}^{2}\frac{3(\ln x)^{2}}{x}dx \) becomes \( \int_{0}^{\ln 2}3u^{2}du \) (since \( \frac{1}{x}dx = du \) and \( (\ln x)^{2}=u^{2} \), and we adjust the limits according to \( u=\ln x \)).

Step3: Integrate

Integrate \( 3u^{2} \) with respect to \( u \). The antiderivative of \( u^{n} \) is \( \frac{u^{n + 1}}{n+1} \), so for \( 3u^{2} \), the antiderivative is \( 3\times\frac{u^{3}}{3}=u^{3} \).

Step4: Evaluate definite integral

Evaluate \( u^{3} \) from \( 0 \) to \( \ln 2 \): \( (\ln 2)^{3}-0^{3}=(\ln 2)^{3} \).

Answer:

\( (\ln 2)^{3} \)