Some Properties Of Definite Integrals

Let I = log (tan x)dx ...(1)
Then, I =




I = -I 2I = 0
[Using e (1)]

We are given the following:
- The equation of the circle: , which represents a circle with radius 2 centered at the origin.
- The line , which passes through the origin and has a slope of .
- We are asked to find the area in the first quadrant bounded by the circle, the line, and the -axis.
Step 1: Find the points of intersection.
We first find the point of intersection of the line and the circle .
Substitute into the equation of the circle: Now substitute into to find : Thus, the point of intersection is . Step 2: Set up the integral for the area. The area we want to calculate is bounded by the circle, the line, and the -axis. The total area in the first quadrant under the circle is given by: where and is the angle subtended by the sector. From the point of intersection , the line makes an angle of with the -axis (since ). Thus, the area of the sector is: Step 3: Subtract the area of the triangle. Next, we calculate the area of the triangle formed by the origin, the point , and the -axis. The area of the triangle is: Step 4: Calculate the desired area. The desired area is the area of the sector minus the area of the triangle: Therefore, the area is square units, and the correct answer is (c).

By analyzing the integral and its relationship with the values of and , we determine that the solution is expressed as .

The integral involves the periodic nature of the sine function, and by splitting the integral at points where the sine function changes its sign, we can evaluate it to get 8.

The greatest integer function gives integer values, and the integral evaluates to 2 after solving by breaking it into intervals where the greatest integer remains constant.

Step 1: Use the Formula for Area.
The area of a loop formed by the curve can be computed using integration. The total area enclosed by the curve is , based on the specific geometry of the curve.
Step 2: Conclusion.
The correct answer is (B), .

Given the functional condition , we use symmetry in the integral to show that the value is .
Step 2: Conclusion.
The correct answer is (B), .

The appearance of in the given equation suggests the substitution: Differentiating: Rewriting the given equation using this substitution: Expanding: Factorizing the terms: This is now a variable separable form: Integrating both sides: Substituting : Comparing with the given general solution, we get: