Compute the definite integral lnx)2 dx Give the exact value.
The Correct Answer and Explanation is:
To compute the definite integral:
I=∫ee3(lnx)2x dxI = \int_{e}^{e^3} \frac{(\ln x)^2}{x} \,dx
we use the substitution method. Let u=lnxu = \ln x, then du=dxxdu = \frac{dx}{x}, transforming the integral into:
I=∫lnelne3u2 duI = \int_{\ln e}^{\ln e^3} u^2 \, du
Since lne=1\ln e = 1 and lne3=3\ln e^3 = 3, the limits of integration adjust accordingly:
I=∫13u2 duI = \int_{1}^{3} u^2 \,du
Now, integrating u2u^2:
∫u2 du=u33\int u^2 \,du = \frac{u^3}{3}
Evaluating this expression from u=1u = 1 to u=3u = 3:
I=[u33]13I = \left[ \frac{u^3}{3} \right]_{1}^{3}
I=333−133I = \frac{3^3}{3} – \frac{1^3}{3}
I=273−13I = \frac{27}{3} – \frac{1}{3}
I=9−13I = 9 – \frac{1}{3}
I=823I = 8 \frac{2}{3}
The exact value of the definite integral is 263\frac{26}{3} or 8 23\frac{2}{3}.
Explanation
The integration process involves recognizing that the given integral can be simplified using substitution. By letting u=lnxu = \ln x, we effectively transform the complicated logarithmic expression into a polynomial integral. This technique streamlines the computation by reducing the complexity associated with logarithmic functions.
After substituting u=lnxu = \ln x and converting dx/xdx/x into dudu, the integral simplifies to ∫u2 du\int u^2 \,du. This transformation enables straightforward polynomial integration. The antiderivative of u2u^2 is determined by the power rule, giving u3/3u^3/3, which is then evaluated at the limits u=1u = 1 and u=3u = 3.
Substituting these values into the antiderivative results in (27/3−1/3)(27/3 – 1/3), which simplifies to 26/326/3 or 8238 \frac{2}{3}. Thus, the definite integral is computed precisely, ensuring accuracy through each step of substitution, integration, and evaluation.
This method is particularly effective for logarithmic integrals where exponentiation is involved, demonstrating the utility of substitution when tackling problems that contain expressions of lnx\ln x.
