The integral of the expression \( t + 2t^3 + 5 \) with respect to \( t \) is:

\[
\int (t + 2t^3 + 5) \, dt
\]

We can integrate term by term:

1. \( \int t \, dt = \frac{t^2}{2} \)
2. \( \int 2t^3 \, dt = 2 \cdot \frac{t^4}{4} = \frac{t^4}{2} \)
3. \( \int 5 \, dt = 5t \)

Now, combining these results:

\[
\frac{t^2}{2} + \frac{t^4}{2} + 5t + C
\]

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

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is the integral of \( t^2 + 3t^3 + 4 \)?
2. How do you find the definite integral of \( t + 2t^3 + 5 \) from \( t = 0 \) to \( t = 1 \)?
3. How do you apply the power rule for integration?
4. How do you integrate \( 3t^2 - 6t + 8 \)?
5. How does the constant of integration affect indefinite integrals?

**Tip:** Always remember to include the constant of integration when computing indefinite integrals!

Explore the step-by-step solution to the integral of the expression t + 2t^3 + 5 with respect to t. This problem covers key calculus concepts, including the power rule for integration and the constant rule. Suitable for students in Grades 10-12.

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