To simplify the equation in the first image in a manner similar to the reference in the second image, let's follow these steps:

1. Identify the structure of the integrals: The first image contains nested integrals with various trigonometric and polynomial terms.

2. Recognize patterns: The second image shows the equation being transformed by identifying common factors and simplifying the expressions inside the integrals.

Let's rewrite and simplify the equation from the first image:

Equation from the First Image

[ \int_0^1 \left[ (x+1) - \left(\frac{x(2x-1)}{x^2} \cos(x)\right) \right] dx + \int_0^1 \left[ (x+1-y) - \left(\frac{x(2x-(1-y))}{x^2} \cos(xy)\right) \right] dx ]

Simplification Steps

1. Simplify Inside the Brackets: Each integral contains two main terms: $(x+1)$ or $(x+1-y)$ and a term involving $\frac{x(2x-\text{some term})}{x^2} \cos(\text{something})$.

2. Simplify Fraction Terms: The fractions $\frac{x(2x-1)}{x^2}$ and $\frac{x(2x-(1-y))}{x^2}$ simplify to $\frac{2x-1}{x}$ and $\frac{2x-(1-y)}{x}$ respectively.

3. Rewrite the Integrals: [ \int_0^1 \left[ (x+1) - \left(\frac{2x-1}{x} \cos(x)\right) \right] dx + \int_0^1 \left[ (x+1-y) - \left(\frac{2x-(1-y)}{x} \cos(xy)\right) \right] dx ]

4. Distribute and Combine Like Terms: Combine and group terms to align with how the second image simplifies them:

   • The $x+1$ or $x+1-y$ terms can be seen as a polynomial part.
   • The cosine term $\cos(x)$ or $\cos(xy)$ contributes a trigonometric component that may be treated separately.

5. Check for Common Factors: Both integrals involve $(x+1)$ like terms and a cosine term. If we follow the reference, we aim to isolate the cosine terms for clearer analysis.

Simplified Form (Based on Reference Style)

[ \int_0^1 \left[ x+1 - \frac{2x-1}{x} \cos(x) \right] dx + \int_0^1 \left[ x+1-y - \frac{2x-(1-y)}{x} \cos(xy) \right] dx ]

Comparison with the Second Image

• The second image uses substitution and common terms to simplify expressions. Each term within the integrals is either a polynomial or a cosine-modified polynomial.
• The simplified equation in the second image has factored common terms, representing a combination of polynomial expressions and cosine-modified expressions, suggesting the possibility of grouping and substitution in the trigonometric components for further simplification.

By carefully aligning terms and recognizing common factors or patterns, the complexity of nested integrals can often be reduced.

Would you like further details on each step, or have any specific questions about this process?

Related Questions

1. How can trigonometric integrals be simplified using substitution techniques?
2. What are common methods to handle nested integrals in calculus?
3. How does recognizing common factors in an equation assist in simplification?
4. Can we apply similar simplification strategies to multi-variable integrals?
5. How do polynomial and trigonometric functions interplay in integral calculus?

Tip

When faced with complex integrals, breaking them down into smaller, more manageable components and looking for common patterns can significantly aid in simplification.