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To perform integration by parts with respect to \( t \) on the given equation, we need to apply the integration by parts formula:

\[
\int u(t)v'(t)\,dt = u(t)v(t) \Big|_{\text{bounds}} - \int u'(t)v(t)\,dt
\]

In the context of the given equation, the integral is:

\[
\int_0^T \int_0^l u(q_n)_t \psi \eta(t) \, dx \, dt
\]

We can identify the parts of the integration:
- \( u(t) = \eta(t) \)
- \( v'(t) = u(q_n)_t \psi \)

Following the formula:
1. Differentiate \( u(t) = \eta(t) \) to get \( u'(t) = \eta'(t) \).
2. Integrate \( v'(t) = u(q_n)_t \psi \) to get \( v(t) = \int u(q_n)_t \psi \, dt \).

Now, applying integration by parts:

\[
\int_0^T u(q_n)_t \psi \eta(t) \, dt = u(q_n)_t \psi \eta(t) \Big|_0^T - \int_0^T u(q_n)_t \psi \eta'(t) \, dt
\]

This yields:

\[
u(q_n)_t \psi \eta(T) - u(q_n)_t \psi \eta(0) - \int_0^T u(q_n)_t \psi \eta'(t) \, dt
\]

Thus, after performing integration by parts with respect to \( t \), the equation becomes:

\[
\int_0^T \int_0^l u(q_n)_t \psi \eta'(t) \, dx \, dt = \text{boundary terms} + \text{integral term with } \eta'(t)
\]

Would you like further details or clarification on this process?

Here are some related questions to expand on the understanding:
1. What are the boundary conditions for \( \eta(t) \) at \( t = 0 \) and \( t = T \)?
2. How would the equation change if \( u(q_n)_t \) is a function of both \( x \) and \( t \)?
3. Can you generalize this process for a different choice of functions \( u(t) \) and \( v'(t) \)?
4. What happens if \( \eta(t) \) is a periodic function?
5. How does integration by parts apply in higher dimensions?

**Tip:** When performing integration by parts in multiple integrals, carefully identify which variable is being integrated and treat others as constants.

请对t做一次分部积分 (Please perform integration by parts with respect to t)

Solve a double integral problem using integration by parts with respect to t. The equation involves u(q_n)_t and psi with double integrals over time and space. Learn how to apply the integration by parts formula step by step.

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