Question:

Step 1: State the property for integrals involving inverse functions. For a monotonically strictly increasing function $f(x)$ over an interval $[a, b]$, the following property holds: $$ \int_a^b f(x)\,dx + \int_{f(a)}^{f(b)} f^{-1}(x)\,dx = b f(b) - a f(a) $$ Step 2: Apply a variation of the inverse function property using the given values. Given $f(1) = 3$ and $f(4) = 5$. The original solution applies an adapted form of the property. $$ \int_3^5 f^{-1}(x)\,dx + \int_3^5 f(x)\,dx = 5 \cdot f(4) - 1 \cdot f(1) = 5(5) - 1(3) = 25 - 3 = 22 $$ This specific value is noted in the original solution but is not directly used in the subsequent steps to calculate the final target integral. The information $\int_3^5 f^2(x)\,dx = 9$ is also provided in the question but is not utilized in the given solution's approach. Step 3: Decompose the target integral into simpler parts. The integral to be evaluated is $2\int_1^4 x(5 - f^{-1}(x))\,dx$. We can split this into two integrals: $$ 2\int_1^4 x(5 - f^{-1}(x))\,dx = 2\left(\int_1^4 5x\,dx - \int_1^4 xf^{-1}(x)\,dx\right) $$ $$ = 10\int_1^4 x\,dx - 2\int_1^4 xf^{-1}(x)\,dx $$ Step 4: Evaluate the first part of the decomposed integral. Calculate the definite integral $10\int_1^4 x\,dx$: $$ 10\int_1^4 x\,dx = 10\left[\frac{x^2}{2}\right]_1^4 $$ $$ = 10\left(\frac{4^2}{2} - \frac{1^2}{2}\right) = 10\left(\frac{16}{2} - \frac{1}{2}\right) = 10\left(\frac{15}{2}\right) = 75 $$ Step 5: Evaluate the second part of the decomposed integral. We need to evaluate $2\int_1^4 xf^{-1}(x)\,dx$. First, consider the indefinite integral $\int xf^{-1}(x)\,dx$. Use integration by parts with $u = f^{-1}(x)$ and $dv = x\,dx$. Then $du = (f^{-1})'(x)\,dx$ and $v = \frac{x^2}{2}$. Applying the integration by parts formula $\int u\,dv = uv - \int v\,du$: $$ \int_1^4 xf^{-1}(x)\,dx = \left[\frac{x^2}{2} f^{-1}(x)\right]_1^4 - \int_1^4 \frac{x^2}{2} (f^{-1})'(x)\,dx $$ The original solution directly states that "Given the constraints, this evaluates to $\frac{16(5) - 1(3)}{2} - 26 = \frac{77}{2} - 26 = \frac{25}{2}$." Following this calculation: The term $\left[\frac{x^2}{2} f^{-1}(x)\right]_1^4$ is evaluated as $\frac{16 \cdot 5 - 1 \cdot 3}{2} = \frac{80 - 3}{2} = \frac{77}{2}$. The remaining integral term is evaluated such that the total sum is $25/2$. $$ \int_1^4 xf^{-1}(x)\,dx = \frac{77}{2} - 26 = \frac{77 - 52}{2} = \frac{25}{2} $$ Therefore, $2\int_1^4 xf^{-1}(x)\,dx = 2 \cdot \frac{25}{2} = 25$. Step 6: Combine the results to find the final answer. Substitute the results from Step 4 and Step 5 back into the decomposed expression from Step 3: $$ 2\int_1^4 x(5 - f^{-1}(x))\,dx = 10\int_1^4 x\,dx - 2\int_1^4 xf^{-1}(x)\,dx $$ $$ = 75 - 25 = 50 $$ The final answer is $\boxed{50}$.