Question

Find the indefinite integral and check your result by differentiation. $$ \int 5 t^{2} d t $$

Short answer

The indefinite integral of \(5t^2\) with respect to \(t\) is \(\frac{5}{3}t^3 + C\).

Step-by-step solution

Apply the Power Rule for Integration

The power rule for integration states that the integral of \(x^n\) with respect to \(x\) is \(\frac{1}{n+1}x^{n+1}\), where \(n\neq-1\). Apply this rule to \(5t^2\) gives \(\int 5t^2 dt = \frac{5}{2+1}t^{2+1} + C = \frac{5}{3}t^3 + C\), where \(C\) is the constant of integration.

Check the Result by Differentiation

Differentiate \(\frac{5}{3}t^3 + C\) with respect to \(t\). According to the power rule for differentiation, this will result in \(5t^2\), which matches the original integrand, confirming that the integration was done correctly.