Question

Find the indefinite integral and check your result by differentiation. $$ \int 6 d x $$

Short answer

The indefinite integral of \(6\) with respect to \(x\) is \(6x + C\), where \(C\) is the constant of integration. The derivative of the obtained result checked for validity, resulted in \(6\), the original function.

Step-by-step solution

Calculation of the indefinite integral

To calculate the indefinite integral of the constant function \(\int 6 dx\), use the rule that the integral of a constant \(c\), denoted by \(\int c dx\), is equal to \(c x + C\), where \(C\) is the constant of integration. Therefore, the antiderivative of \(6\) is \(6x + C\).

Checking the result by differentiation

Differentiate the obtained result \(6x + C\) with respect to \(x\) to verify the accuracy of the solution. By applying the power rule, which states that the derivative of \(x^n\), denoted by \(d/dx[x^n] = n x^{n - 1}\), and noting that the derivative of any constant is zero, you find the derivative of \(6x + C\) to be \(6\).