https://file.aiquickdraw.com/mathbot/file/17333385249000ig8zt8n.jpg

The integral in the image is:

\[
\int_{0}^{\ln(2)} \cosh(x) \sin(\pi \sinh(x)) \, dx
\]

To solve this, we proceed as follows:

### Step 1: Substitution
Let:
\[
u = \sinh(x)
\]
Then:
\[
\cosh(x) \, dx = du
\]

When \(x = 0\), \(u = \sinh(0) = 0\).  
When \(x = \ln(2)\), \(u = \sinh(\ln(2))\).

### Step 2: Change of limits and integral
The integral becomes:
\[
\int_{0}^{\sinh(\ln(2))} \sin(\pi u) \, du
\]

### Step 3: Solve the integral
The integral of \(\sin(\pi u)\) is:
\[
\int \sin(\pi u) \, du = -\frac{1}{\pi} \cos(\pi u)
\]

Using the limits \(u = 0\) and \(u = \sinh(\ln(2))\), the evaluation becomes:
\[
-\frac{1}{\pi} \cos(\pi u) \Big|_{0}^{\sinh(\ln(2))}
\]

### Step 4: Simplify
Substitute the limits:
\[
-\frac{1}{\pi} \left[\cos(\pi \sinh(\ln(2))) - \cos(0)\right]
\]

Since \(\cos(0) = 1\), this simplifies to:
\[
-\frac{1}{\pi} \left[\cos(\pi \sinh(\ln(2))) - 1\right]
\]

Thus, the final result is:
\[
\frac{1}{\pi} \left[1 - \cos(\pi \sinh(\ln(2)))\right]
\]

Let me know if you'd like further clarifications or a detailed breakdown! Here are some related questions for exploration:

1. What is the hyperbolic sine (\(\sinh(x)\)) and hyperbolic cosine (\(\cosh(x)\)) function?
2. How do substitutions simplify integral evaluations?
3. What are the geometric interpretations of hyperbolic functions?
4. Why does the integral of \(\sin(ax)\) include a factor of \(1/a\)?
5. How do definite integrals work with trigonometric and hyperbolic substitutions?

**Tip:** When working with trigonometric and hyperbolic functions, always double-check their domains and relationships for accurate substitution.

Evaluate the definite integral \( \int_{0}^{\ln(2)} \cosh(x) \sin(\pi \sinh(x)) \, dx \).

Explore a detailed solution to the definite integral \( \int_{0}^{\ln(2)} \cosh(x) \sin(\pi \sinh(x)) \, dx \). Learn about hyperbolic and trigonometric functions, substitution techniques, and evaluation of definite integrals. Suitable for advanced 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