Question

Find the indefinite integral and check your result by differentiation. $$ \int(5-x) d x $$

Short answer

The indefinite integral of \(5-x\) is \(5x - \frac{x^2}{2} + C\).

Step-by-step solution

Integration

Take the integral of the function \(5-x\) with respect to \(x\). It's important to remember that the integral of a constant \(c\) is \(cx\) and the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\). So, in this case, integrating \(5\) with respect to \(x\) yields \(5x\), and integrating \(-x\) yields \(-\frac{x^2}{2}\). Adding these two results gives the indefinite integral of \(5-x\), which is \(5x - \frac{x^2}{2} + C\), where \(C\) is the constant of integration.

Checking the result by differentiation

Now, differentiate the result of the indefinite integral, which is \(5x - \frac{x^2}{2} + C\), with respect to \(x\). The derivative of \(5x\) is \(5\), the derivative of \(-\frac {x^2}{2}\) is \(-x\) and the derivative of the constant \(C\) is \(0\). Adding these derivatives together gives the original function, \(5 - x\). This verifies that the integral was computed correctly.