`int (3 t^(2) + 5) dt` :

Options

A. `t^(3) + 5t`
B. 6t
C. 6t + c
D. `t^(3) + 5t + c`

Answer

To solve the integral \( \int (3t^2 + 5) \, dt \), we will follow these steps:

### Step 1: Set up the integral
Let \( I = \int (3t^2 + 5) \, dt \).

### Step 2: Split the integral
We can separate the integral into two parts:
\[ I = \int 3t^2 \, dt + \int 5 \, dt \]

### Step 3: Factor out constants
Since 3 and 5 are constants, we can factor them out of the integrals:
\[ I = 3 \int t^2 \, dt + 5 \int dt \]

### Step 4: Integrate each part
Now, we will integrate each part separately.

1. For \( \int t^2 \, dt \):
Using the power rule of integration, \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \):
\[ \int t^2 \, dt = \frac{t^{2+1}}{2+1} = \frac{t^3}{3} \]

2. For \( \int dt \):
\[ \int dt = t \]

### Step 5: Substitute back into the equation
Now, substituting these results back into our expression for \( I \):
\[ I = 3 \left( \frac{t^3}{3} \right) + 5t \]

### Step 6: Simplify the expression
The \( 3 \) in the numerator and denominator cancels out:
\[ I = t^3 + 5t \]

### Step 7: Add the constant of integration
Finally, we add the constant of integration \( C \):
\[ I = t^3 + 5t + C \]

Thus, the final answer is:
\[ \int (3t^2 + 5) \, dt = t^3 + 5t + C \]

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