Question

Suppose that the function \(f, g, f^{\prime}\) and \(g^{\prime}\) are continuous over \([0,1], \mathrm{g}(\mathrm{x}) \neq 0\) for \(\mathrm{x} \in[0,1]\), \(\mathrm{f}(0)=0, \mathrm{~g}(0)=\pi, \mathrm{f}(1)=\frac{2009}{2}\) and \(\mathrm{g}(1)=1\). Find the value of the definite integral, \(\int_{0}^{1} \frac{\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}^{\prime}(\mathrm{x})\left\\{\mathrm{g}^{2}(\mathrm{x})-1\right\\}+\mathrm{f}^{\prime}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})\left\\{\mathrm{g}^{2}(\mathrm{x})+1\right\\}}{\mathrm{g}^{2}(\mathrm{x})} \mathrm{dx}\)

Short answer

Answer: The value of the definite integral is \(\frac{2009}{2}\).

Step-by-step solution

Introduce the function h(x)

Let's define a new function, h(x), by \(h(\mathrm{x})=\mathrm{f}(\mathrm{x})\cdot\mathrm{g}(\mathrm{x})\).

Differentiate h(x)

Using the product rule, find the derivative of h(x): \(h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)\).

Rearrange the expression for h'(x) and factor

Arrange the terms as follows: \(h^{\prime}(x)=\frac{f^{\prime}(x)g(x)(g^2(x)+1)+f(x)g^{\prime}(x)(g^2(x)-1)}{g^2(x)}\). Notice that this expression is precisely the integrand in the desired definite integral.

Integrate the expression

We integrate the expression obtained in Step 3 from 0 to 1: \(\int_{0}^{1} h^{\prime}(x) \, dx= \int_{0}^{1} \frac{f^{\prime}(x)g(x)(g^2(x)+1)+f(x)g^{\prime}(x)(g^2(x)-1)}{g^2(x)} \, dx\).

Apply the Fundamental Theorem of Calculus

Since we have integrated the derivative of h(x), we can use the Fundamental Theorem of Calculus to find the value of the definite integral: \(h(1)-h(0)=\int_{0}^{1} h^{\prime}(x) \, dx\).

Evaluate h(1) and h(0)

Remember that \(h(\mathrm{x})=\mathrm{f}(\mathrm{x})\cdot\mathrm{g}(\mathrm{x})\). Using this definition and the given conditions for f and g, we can find the values of h(1) and h(0): \(h(0)=f(0)\cdot g(0)=0\cdot \pi=0\) \(h(1)=f(1)\cdot g(1)=\frac{2009}{2}\cdot 1=\frac{2009}{2}\).

Calculate the value of the definite integral

Substitute the values of h(1) and h(0) back into the expression obtained in Step 5: \(\int_{0}^{1} \frac{f(x)g^{\prime}(x)(g^2(x)-1)+f^{\prime}(x)g(x)(g^2(x)+1)}{g^2(x)} \, dx = h(1)-h(0)=\frac{2009}{2}-0=\frac{2009}{2}\). The value of the definite integral is \(\frac{2009}{2}\).