Integration (sinx+cosx+tanx+cosecx+secx+Cotx)dx=? Plzzz give me the and with appropriate steps .....here integration reffers to the sign of integration.

To tackle the integral of the expression sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x), we’ll break it down step by step. The integral can be quite complex due to the variety of trigonometric functions involved. However, by systematically approaching each part, we can find the solution.

The Integral Expression

We are looking to integrate the function:

∫ (sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x)) dx

Breaking Down the Integral

This integral can be separated into individual components for easier handling:

• ∫ sin(x) dx
• ∫ cos(x) dx
• ∫ tan(x) dx
• ∫ csc(x) dx
• ∫ sec(x) dx
• ∫ cot(x) dx

Integrating Each Component

Let’s solve each integral one by one:

1. Integral of sin(x)

∫ sin(x) dx = -cos(x) + C₁

2. Integral of cos(x)

∫ cos(x) dx = sin(x) + C₂

3. Integral of tan(x)

∫ tan(x) dx = -ln|cos(x)| + C₃

4. Integral of csc(x)

∫ csc(x) dx = -ln|csc(x) + cot(x)| + C₄

5. Integral of sec(x)

∫ sec(x) dx = ln|sec(x) + tan(x)| + C₅

6. Integral of cot(x)

∫ cot(x) dx = ln|sin(x)| + C₆

Combining the Results

Now, we can combine all these results into a single expression. Since the constants of integration (C₁, C₂, etc.) will combine into a single arbitrary constant (C), we can express the final integrated result as:

∫ (sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x)) dx = -cos(x) + sin(x) - ln|cos(x)| - ln|csc(x) + cot(x)| + ln|sec(x) + tan(x)| + ln|sin(x)| + C

Final Result

Thus, the complete integral can be neatly summarized as:

∫ (sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x)) dx = -cos(x) + sin(x) - ln|cos(x)| - ln|csc(x) + cot(x)| + ln|sec(x) + tan(x)| + ln|sin(x)| + C

Remember, the constant C represents the constant of integration, which can take any value. This integration exercise illustrates the power of breaking down complex expressions into manageable parts, making the problem more approachable.