We are asked to evaluate the definite integral:

$\int_{-7}^{7} (\sinh(x) + \cosh(x)) \, dx$

Step 1: Analyze the integrand

The integrand is the sum of the hyperbolic sine and hyperbolic cosine functions:

$\sinh(x) + \cosh(x)$

Now, let's recall the properties of these functions:

• $\sinh(x)$ is an odd function, meaning $\sinh(-x) = -\sinh(x)$.
• $\cosh(x)$ is an even function, meaning $\cosh(-x) = \cosh(x)$.

Step 2: Break the integral into two parts

We can split the given integral into two simpler integrals:

$\int_{-7}^{7} (\sinh(x) + \cosh(x)) \, dx = \int_{-7}^{7} \sinh(x) \, dx + \int_{-7}^{7} \cosh(x) \, dx$

Step 3: Evaluate each integral

• The first integral, $\int_{-7}^{7} \sinh(x) \, dx$, is zero because $\sinh(x)$ is an odd function and the integral of an odd function over a symmetric interval (from $-a$ to $a$) is always zero. Thus,

$\int_{-7}^{7} \sinh(x) \, dx = 0$

• The second integral, $\int_{-7}^{7} \cosh(x) \, dx$, can be simplified by using the fact that $\cosh(x)$ is an even function. This allows us to double the integral from 0 to 7:

$\int_{-7}^{7} \cosh(x) \, dx = 2 \int_0^{7} \cosh(x) \, dx$

The integral of $\cosh(x)$ is straightforward:

$\int \cosh(x) \, dx = \sinh(x)$

So,

$2 \int_0^{7} \cosh(x) \, dx = 2 [\sinh(x)]_0^7 = 2 (\sinh(7) - \sinh(0))$

Since $\sinh(0) = 0$, this simplifies to:

$2 (\sinh(7) - 0) = 2 \sinh(7)$

Final answer:

$\int_{-7}^{7} (\sinh(x) + \cosh(x)) \, dx = 2 \sinh(7)$

Would you like more details or have any questions?

Here are 5 related questions you may find useful:

1. What are the key differences between hyperbolic and trigonometric functions?
2. How can we apply symmetry in integrals of even and odd functions?
3. Can you explain why $\sinh(x)$ is an odd function and $\cosh(x)$ is even?
4. What are some real-world applications of hyperbolic functions in physics or engineering?
5. How would this integral change if the limits were not symmetric?

Tip: Remember, odd functions integrated over symmetric limits around 0 always yield 0, which can simplify many integral evaluations.