Chapter 7.3, Problem 36E

To determine

To verify whether ${\displaystyle \int xg(x)dx}=xf(x)-{\displaystyle \int f(x)dx}$ under the condition $f^{\prime }(x)=g(x)$

Answer to Problem 36E

The given statement is true under the said condition

Explanation of Solution

Given information: If $f^{\prime }(x)=g(x)$

Formula used: ${\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int vdx}\right)}}}dx$

Calculation :

Given $f^{\prime }(x)=g(x)$

Multiply both sides to x and integrating, the expression becomes as:

  ${\displaystyle \int xg(x)}dx={\displaystyle \int x{f}^{\prime }(x)}dx$

  $=xf(x)-{\displaystyle \int f(x)}dx$

Hence, the expression is true.