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### Question 17: Even function identification
We are asked to identify which of the given functions is even.

- **A function \( f(x) \) is even if**: \( f(-x) = f(x) \) for all \( x \).

Let’s evaluate each function:
- \( y = \tan x \): The tangent function is **odd**, not even, because \( \tan(-x) = -\tan(x) \).
- \( y = e^x \): The exponential function is **neither even nor odd**, because \( e^{-x} \neq e^x \).
- \( y = x^2 + x^4 \): This is **even**, since both \( x^2 \) and \( x^4 \) are even powers, and \( (-x)^2 + (-x)^4 = x^2 + x^4 \).
- \( y = \sin x \): The sine function is **odd**, because \( \sin(-x) = -\sin(x) \).
- \( y = x + x^3 \): This is **odd**, since \( (-x) + (-x)^3 = -x - x^3 \).

Thus, the correct answer is **\( y = x^2 + x^4 \)**.

### Question 18: Definite Integral
We are asked to compute the definite integral
\[
\int_{-3}^{3} \left( 1 + x + x^3 + x^5 + x^7 \right) dx.
\]

The key observation here is that the integrand consists of a mixture of even and odd powers of \(x\). Specifically:
- \( 1 \) is even.
- \( x \) is odd.
- \( x^3 \), \( x^5 \), and \( x^7 \) are odd.

**Property of definite integrals for odd and even functions:**
- The integral of an **odd function** over a symmetric interval \( [-a, a] \) is **0**.
- The integral of an **even function** can be calculated normally.

Thus, the terms \( x \), \( x^3 \), \( x^5 \), and \( x^7 \) will contribute **0** to the integral, and we only need to integrate the constant term \( 1 \), which is even:
\[
\int_{-3}^{3} 1 \, dx = [x]_{-3}^{3} = 3 - (-3) = 6.
\]

The correct answer is **6**.

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Would you like any additional details or explanations on these problems?

Here are five related questions for further practice:
1. How do you determine if a function is odd?
2. What is the integral of \( x^2 \) over a symmetric interval \( [-a, a] \)?
3. How do you evaluate the integral of an odd function over any general interval?
4. What is the definition of an even function in terms of symmetry?
5. How can you decompose a complex function into even and odd parts?

**Tip:** When calculating definite integrals over symmetric intervals, always check if the function is odd or even to potentially simplify the calculation!

Which one of the following functions is even? And Find the definite integral \( \int_{-3}^{3} \left( 1 + x + x^3 + x^5 + x^7 \right) dx \).

Solve the problem of identifying even functions and computing the definite integral \( \int_{-3}^{3} \left( 1 + x + x^3 + x^5 + x^7 \right) dx \) by applying properties of odd and even functions. Learn about integral properties and their applications to function symmetry.

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