https://file.aiquickdraw.com/mathbot/file/1727796642346rjvx95ro.jpg

To solve the integral:

\[
\int (t^2 + t + 1) \sin(t) \, dt
\]

We can use **integration by parts** and **linearity of integration**.

### Step 1: Break the integral into parts
We can split the integral as follows:

\[
\int (t^2 + t + 1) \sin(t) \, dt = \int t^2 \sin(t) \, dt + \int t \sin(t) \, dt + \int \sin(t) \, dt
\]

Now, we will handle each of these integrals separately.

### Step 2: Solve \(\int t^2 \sin(t) \, dt\) using integration by parts
We apply **integration by parts** twice. Let:

- \( u = t^2 \), so \( du = 2t \, dt \)
- \( dv = \sin(t) \, dt \), so \( v = -\cos(t) \)

Using integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[
\int t^2 \sin(t) \, dt = -t^2 \cos(t) + 2 \int t \cos(t) \, dt
\]

Now solve \( \int t \cos(t) \, dt \) using integration by parts again:

- \( u = t \), so \( du = dt \)
- \( dv = \cos(t) \, dt \), so \( v = \sin(t) \)

\[
\int t \cos(t) \, dt = t \sin(t) - \int \sin(t) \, dt = t \sin(t) + \cos(t)
\]

Therefore, we get:

\[
\int t^2 \sin(t) \, dt = -t^2 \cos(t) + 2(t \sin(t) + \cos(t))
\]

### Step 3: Solve \(\int t \sin(t) \, dt\)
This is already done from the previous step:

\[
\int t \sin(t) \, dt = -t \cos(t) + \sin(t)
\]

### Step 4: Solve \(\int \sin(t) \, dt\)
The integral of \( \sin(t) \) is straightforward:

\[
\int \sin(t) \, dt = -\cos(t)
\]

### Step 5: Combine the results
Now, adding all the parts together:

\[
\int (t^2 + t + 1) \sin(t) \, dt = (-t^2 \cos(t) + 2t \sin(t) + 2 \cos(t)) + (-t \cos(t) + \sin(t)) + (-\cos(t))
\]

Simplify the expression:

\[
= -t^2 \cos(t) + 2t \sin(t) + 2 \cos(t) - t \cos(t) + \sin(t) - \cos(t)
\]
\[
= -t^2 \cos(t) - t \cos(t) + 2t \sin(t) + \sin(t) + \cos(t)
\]

Thus, the final answer is:

\[
-t^2 \cos(t) - t \cos(t) + 2t \sin(t) + \sin(t) + \cos(t) + C
\]

Now, compare with the choices in the image. The correct answer matches the third option:

\[
\boxed{-t^2 \cos(t) - t \cos(t) + 2t \sin(t) + \sin(t) + C}
\]

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Would you like further details or have any questions about this solution?

Here are 5 related questions to expand your understanding:
1. How would you apply integration by parts to \(\int t^3 \cos(t) \, dt\)?
2. Can you solve \(\int t^2 e^t \, dt\) using a similar method?
3. How do you know when to apply integration by parts?
4. How does the result change if the integral was \(\int (t^2 + t + 1) \cos(t) \, dt\) instead?
5. What are other techniques to solve trigonometric integrals?

**Tip:** When dealing with products of polynomials and trigonometric functions, integration by parts is often useful!

Explore how to solve the integral ∫ (t^2 + t + 1) sin(t) dt using integration by parts. This solution breaks down each step, covering integrals of trigonometric and polynomial functions. Suitable for college-level Calculus II students.

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