Integration

Step 1: Understand the given integral form
The given integral is: Factorizing the denominator, Thus, the given integral simplifies into the sum of two inverse trigonometric functions.

Step 2: Evaluate the constants
From the given formula, Comparing both sides, we analyze: - and are expressions derived from the decomposition. - is a constant to be determined.

Step 3: Compute the required values
Evaluating the given expression: Using standard values from inverse trigonometric functions, solving step by step gives:

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Simplify the Integrand
We start with: Rewriting: Thus, Since is periodic, we consider the periodicity of the function over the given interval.

Step 2: Evaluate Over One Period
The function inside the square root is periodic with period . Hence, we analyze its integral over and then scale it for . Over , the integral evaluates to a known result: Since corresponds to full cycles of , we multiply:

Step 3: Compute the Given Expression
Thus, the final value is:

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Identifying the Order of the Differential Equation
The order of a differential equation is the highest order derivative present in the equation. Here, the highest order derivative is , which means the order of the differential equation is:

Step 2: Identifying the Degree of the Differential Equation
The degree of a differential equation is defined as the highest exponent of the highest order derivative after removing radicals and fractions. In this equation, the term contains a fractional power. To determine the degree, we must first eliminate this radical by squaring both sides. After squaring, the highest exponent of (the highest order derivative) is found to be:

Step 3: Conclusion
Thus, the correct answer is:

Step 1: Identifying the Given Differential Equation
The given general solution is: Differentiating both sides: From the given differential equation: Substituting : Rearrange:

Step 2: Finding Integrating Factor (IF)
A standard linear differential equation is of the form: The integrating factor (IF) is given by: From the given form, we identify: Thus, the integrating factor is:

Step 3: Conclusion
Thus, the integrating factor is:

Step 1: Identify the Integral Form
Given: Observing the structure, we let: Differentiating: Thus, the integral transforms into a standard exponential integral form:

Step 2: Solve the Integral
The standard result is: Thus,

Step 3: Compute

Step 4: Conclusion
Thus, the correct answer is:

Step 1: Set up the definite integral for the area.
The area of the region bounded by a curve , the x-axis, and the lines and is given by . Here, , , and .

Step 2: Split the integral based on the sign of the function.
The function is negative for and positive for . We need to split the integral at . The integral becomes:

Step 3: Evaluate the first integral.


Step 4: Evaluate the second integral.


Step 5: Calculate the total area.
The total area is the sum of the areas from the two parts.