Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. $$ \int(1+3 t) t^{2} d t $$

Step-by-Step Solution

Answer

The indefinite integral of the function is \( \frac{1}{3}t^3 + \frac{3}{4}t^4 + C \)

Step 1: Identify the function within the integral

The function within the integral sign in this case is \((1+3t)t^{2}\). There is an implicit multiplication between \(t^{2}\) and the expression in the parentheses \(1+3t\). This means that the function can be expanded.

Step 2: Expand the expression within the integral

By expanding the expression, the function becomes \(t^{2} + 3t^{3}\). The function can now be broken down and each term can be integrated separately.

Step 3: Use the Power Rule for Integration

The power rule for integration states that the integral of \(x^n\), with respect to \(x\), where \(n\) is any real number except -1, is \(\frac{1}{n+1}x^{n+1}\). Apply this rule to each term of the function to find the antiderivative. This gives \( \frac{1}{3}t^3 + \frac{3}{4}t^4 \).