Problem 29 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a differentiable function. If \(F\) is an antiderivative of \(f\) on \([a, b]\), then show that $$ \int_{a}^{b} f^{2}(x) d x=F(b) F^{\prime}(b)-F(a) F^{\prime}(a)-\int_{a}^{b} F(x) F^{\prime \prime}(x) d x $$

Step-by-Step Solution

Answer

The short version of the answer would be: To prove the given integral relationship, we apply integration by parts with \(u = F(x)\) and \(dv = F'(x)dx\). We then substitute the function \(f(x)\) with its antiderivative \(F(x)\), resulting in the integral \(\int_{a}^{b} F^{\prime 2}(x) \ dx\). After applying the integration by parts formula and evaluating the integral over the interval \([a, b]\), we obtain the desired result: \[ \int_{a}^{b} f^2(x) \ dx = F(b)F'(b) - F(a)F'(a) - \int_{a}^{b} F(x)F''(x) \ dx \]

Step 1: Apply the integration by parts formula

We will use the formula for integration by parts, which is given by \[ \int u \ dv = uv - \int v \ du \] To choose the appropriate 'u' and 'dv' for our integral, we need to define \(u = F(x)\), yielding \(du = F'(x)dx\). Then we set \(dv = F'(x)dx\) and upon integration, obtain \(v = F(x)\). Now, let's apply the formula to our integral.

Step 2: Set up the integral and substitute

We want to compute the integral of \(f^2(x)\), so let's substitute the relationship between \(f(x)\) and \(F(x)\), i.e., \(f(x) = F'(x)\). Our integral becomes: \[ \int_{a}^{b} F^{\prime 2}(x) \ dx \] Now, we can apply integration by parts with \(u = F(x)\) and \(dv = F'(x)dx\), which we defined earlier: \begin{align*} \int_{a}^{b} F^{\prime 2}(x) \ dx &= \int_{a}^{b} F'(x) \ F'(x) \ dx \\ &= \int_{a}^{b} u \ dv \end{align*}

Step 3: Apply the integration by parts formula

Next, plug in the 'u', 'v', 'du', and 'dv' into the integration by parts formula: \[ uv - \int_{a}^{b} v \ du = F(x)F'(x) - \int_{a}^{b} F(x)F''(x) \ dx \]

Step 4: Evaluate the integral

Now we need to evaluate the integral over the interval \([a, b]\). Apply the limits of integration to \(uv\): \[ \left. F(x)F'(x) \right|_{a}^{b} = F(b)F'(b) - F(a)F'(a) \]

Step 5: Combine the terms and rewrite the results

We have now evaluated all the terms in the integration by parts formula. Combining them, we get: \[ \int_{a}^{b} f^2(x) \ dx = F(b)F'(b) - F(a)F'(a) - \int_{a}^{b} F(x)F''(x) \ dx \] This is the required result.