Compute the integral 0 π⟨cos(2t),tet,t2⟩dt.

To compute the integral of the vector function \(\langle \cos(2t), te^t, t^2 \rangle\) from 0 to \(\pi\), we will integrate each component of the vector function separately over the interval \([0, \pi]\).

The integral is:

\[
\int_0^\pi \langle \cos(2t), te^t, t^2 \rangle dt
= \langle \int_0^\pi \cos(2t) dt, \int_0^\pi te^t dt, \int_0^\pi t^2 dt \rangle
\]

### 1. First component: \(\int_0^\pi \cos(2t) dt\)
Using the standard integral of cosine:
\[
\int \cos(2t) dt = \frac{1}{2} \sin(2t)
\]
Evaluating this from 0 to \(\pi\):
\[
\left[ \frac{1}{2} \sin(2t) \right]_0^\pi = \frac{1}{2} (\sin(2\pi) - \sin(0)) = \frac{1}{2} (0 - 0) = 0
\]

### 2. Second component: \(\int_0^\pi te^t dt\)
This integral requires integration by parts.

Let:
\[
u = t \quad \text{and} \quad dv = e^t dt
\]
Then:
\[
du = dt \quad \text{and} \quad v = e^t
\]
Using integration by parts:
\[
\int te^t dt = te^t - \int e^t dt = te^t - e^t
\]
Evaluating from 0 to \(\pi\):
\[
\left[ te^t - e^t \right]_0^\pi = \left( \pi e^\pi - e^\pi \right) - \left( 0e^0 - e^0 \right) = \pi e^\pi - e^\pi + 1
\]
Thus, the second component is:
\[
(\pi - 1) e^\pi + 1
\]

### 3. Third component: \(\int_0^\pi t^2 dt\)
This is a standard power rule integral:
\[
\int t^2 dt = \frac{t^3}{3}
\]
Evaluating from 0 to \(\pi\):
\[
\left[ \frac{t^3}{3} \right]_0^\pi = \frac{\pi^3}{3} - 0 = \frac{\pi^3}{3}
\]

### Final Answer
Combining the results from the three components, the integral of the vector function is:

\[
\int_0^\pi \langle \cos(2t), te^t, t^2 \rangle dt = \langle 0, (\pi - 1) e^\pi + 1, \frac{\pi^3}{3} \rangle
\]

Would you like more details on any specific step or component? 

Here are 5 related questions you might find helpful:
1. How do you apply integration by parts to more complex functions?
2. What are the standard trigonometric integrals and their uses?
3. How can vector-valued functions be integrated in higher dimensions?
4. What are some applications of integrating vector functions?
5. How do boundary conditions affect the results of definite integrals?

**Tip:** Remember that for vector integrals, integrating each component separately simplifies the computation.

Explore the step-by-step solution to compute the integral of the vector function ⟨cos(2t), te^t, t^2⟩ over the interval [0, π]. This solution uses key calculus concepts like integration by parts, the power rule, and trigonometric integrals, suitable for undergraduate calculus students.

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