Questions: Evaluate the integral. [ int(4 t^3+6) t^2 d t=square ] (Type an exact answer. Use parentheses to clearly denote the argument of each function.)

Solution Steps

To evaluate the integral \(\int (4t^3 + 6) t^2 \, dt\), we first simplify the integrand by distributing \(t^2\) inside the parentheses. Then, we integrate each term separately using the power rule for integration.

We start with the integral

\[ \int (4t^3 + 6) t^2 \, dt. \]

Distributing \(t^2\) gives us

\[ \int (4t^5 + 6t^2) \, dt. \]

Next, we apply the power rule for integration to each term:

1. For \(4t^5\), the integral is

\[ \frac{4}{6} t^6 = \frac{2}{3} t^6. \]

2. For \(6t^2\), the integral is

\[ \frac{6}{3} t^3 = 2t^3. \]

Combining these results, we have:

\[ \int (4t^5 + 6t^2) \, dt = \frac{2}{3} t^6 + 2t^3 + C, \]

where \(C\) is the constant of integration.

Final Answer