Question

You should solve the following problems without using a graphing calculator. True or False If \(f^{\prime}(x)=g(x),\) then \(\int x g(x) d x=\) \(x f(x)-\int f(x) d x .\) Justify your answer.

Short answer

The statement is True. This can be demonstrated by applying the formula for Integration by Parts.

Step-by-step solution

Recall the Formula for Integration by Parts

The formula for integration by parts is \(\int u dv = uv - \int v du \), where \(u\) and \(dv\) are parts of the function being integrated which we have to choose. Normally, we let \(u\) be a function that simplifies when we differentiate it, and \(dv\) be a function that we can easily integrate.

Apply the Definition in The Problem

Here, if we let \(u = x\) and \(dv = g(x) dx = f'(x) dx\), then \(du = dx\) and \(v = f(x)\). Following the formula for integration by parts, we get \(\int x g(x) dx = \int x f'(x) dx = x f(x) - \int f(x) dx\).

Making the Conclusion

Thus, the statement is true, as we found that \(\int x g(x) dx = x f(x) - \int f(x) dx\) when \(g(x) = f'(x)\), by applying the Integration by Parts formula.