Question

Find the indefinite integral and check your result by differentiation. $$ \int 5 x^{-3} d x $$

Short answer

The indefinite integral of \(5x^{-3}\) is \(-\frac{5}{2} x^{-2} + C\), and the differentiation of this result confirms the initial function \(-5x^{-3}\)

Step-by-step solution

Apply Power Rule of Integration

The power rule states that the integral of \(x^n\), where n ≠ -1, is \(\frac{x^{n+1}}{n+1} + C\), where C is the constant of integration. Apply this rule to our specific problem, we get: \( \int 5x^{-3} dx = 5 \int x^{-3} dx = 5 (\frac{x^{-2}}{-2}) + C \)

Simplify the Result

Next, we should simplify the expression. The result can be rewritten as: \( -\frac{5}{2} x^{-2} + C \)

Check Result by Differentiating

Now, differentiate the result to check the solution. Use the power rule for differentiation: \(f(x) = x^n\)' = \(nx^{n-1}\). Applying this rule, we get: \((-\frac{5}{2} x^{-2} + C)' = -5x^{-3} + 0 = -5x^{-3}\)