Question

Suppose that \(f(1) = 2,f(4) = 7,f'(1) = 5,f'(4) = 3,\) and is continuous. Find the value of

Short answer

Apply by parts integral formula to the given definite integral

Apply by parts integration to the given integrand, then the final answer can be obtained by using specified values to the general solution of the general solution of the given definite integral

Step-by-step solution

Step-1: Given Data

It is given that \(f(1) = 2,f(4) = 7,f'(1) = 5\) and \(f'(4) = 3\)

Also, it is said that is continuous

Let us first determine the indefinite integral as by using by parts integration formula

Now,

Here ‘c’ is an arbitrary constant

Now let us consider \(u = x\) and \(v = f'(x)\)

Now \(du = dx\) and \(dv = f'(x)dx \Rightarrow v = f'(x)\)

Step-2: Finding the Integral

Now applying the limits and then the definite integers

Will give the solution using the given values and the Leibnitz formula:

\( = (4)f'(4) - f(4) - \left( {(1)f'(1) - f(1)} \right)\)

Now on putting the specified values, we get

\( = 4 \times 3 - 7 - \left( {(1)(5) - 2} \right)\)

\( = 12 - 7 - 3\)

\( = 2\)

Hence,

The value of the integral is 2