https://file.aiquickdraw.com/mathbot/file/17289733750266pd7daeb.jpg

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.

Evaluate the definite integral: ∫ from -7 to 7 (sinh(x) + cosh(x)) dx

Explore the step-by-step evaluation of the definite integral ∫ from -7 to 7 (sinh(x) + cosh(x)) dx. Learn about hyperbolic functions, odd and even function properties, and the use of symmetry in integral calculations. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation