Question

Suppose that \(f(1)=2, f(4)=7, f^{\prime}(1)=5, f^{\prime}(4)=3\) and \(\mathrm{f}^{\prime \prime}\) is continuous, Find the value of \(\int_{1}^{4} x f^{\prime \prime}(x) d x\)

Short answer

Question: Find the value of the integral \(\int_{1}^{4} x f^{\prime \prime}(x) d x\) given \(f(1)=2\), \(f(4)=7\), \(f^{\prime}(1)=5\), \(f^{\prime}(4)=3\), and \(f^{\prime\prime}(x)\) is continuous. Answer: The value of the integral \(\int_{1}^{4} x f^{\prime\prime}(x) d x = 2\).

Step-by-step solution

Identify the parts for integration by parts

As the problem requires the evaluation of the integral \(\int_{1}^{4} x f^{\prime \prime}(x) d x\), we can resort to integration by parts to find it. Integration by parts uses the formula: \[\int {u d v}=uv - \int {vd u}\] For this integral, let's choose \(u = x\) and \(dv = f^{\prime \prime}(x)dx\).

Calculate du and v

Once we have our u and dv, we need to determine du and v. According to our selections: \(du = dx\) and to find \(v\), we integrate \(dv = f^{\prime \prime}(x)dx\) with respect to \(x\). Since \(v\) is the antiderivative of \(f^{\prime\prime}(x)\), we have \(v = f^{\prime}(x) + C\), where C is the constant of integration.

Apply integration by parts formula

Applying the integration by parts formula, we now have: \[\int_{1}^{4} x f^{\prime \prime}(x) d x = (x f^{\prime}(x))\Big|_{1}^{4} - \int_{1}^{4} f^{\prime}(x) d x\]

Evaluate the definite integral

Now, we need to evaluate the definite integral. Using the provided values of \(f^{\prime}(1)\) and \(f^{\prime}(4)\), we get: \[(4f^{\prime}(4) - f^{\prime}(1)) - (F(4) - F(1)) = (4(3) - (5)) - (7 - 2) = 7 - 5 = 2\] So, the value of \(\int_{1}^{4} x f^{\prime\prime}(x) d x = 2\).