Some Properties Of Definite Integrals

Step 1: Understanding the Concept:
This problem requires evaluating a definite integral involving an absolute value function. The key is to split the integral at the point where the expression inside the absolute value changes sign.
Step 2: Key Formula or Approach:
The definition of the absolute value function is:
We use the property of definite integrals: .
Step 3: Detailed Explanation:
The expression inside the absolute value is . This expression is zero when x = 2.


When , is negative, so .
When , is non-negative, so .
Since the point x = 2 is within our integration interval [1, 4], we split the integral at x = 2:
Now substitute the appropriate expression for in each integral:
Evaluate the first integral:
Evaluate the second integral:
Add the results of the two integrals:
Step 4: Final Answer:
The value of the integral is }.

Step 1: Understanding the Concept:
We need to evaluate four different definite integrals and match them with their correct values from List-II. This involves using various integration techniques such as substitution, properties of odd/even functions, and partial fractions.
Step 3: Detailed Explanation:


(A) :} Use substitution. Let . Then .
When , . When , .
The integral becomes . This matches (III).

(B) :} Let . Check if it's an odd or even function.
.
Since , the function is odd. The integral of an odd function over a symmetric interval is 0. This matches (IV).

(C) :} The antiderivative of is .
. This matches (I).

(D) :} Use partial fraction decomposition: .
The integral is .
. This matches (II).

Step 4: Final Answer:
The correct matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II). This corresponds to option (4).

Step 1: Understanding the Concept:
This definite integral can be solved efficiently using a property of definite integrals, often known as the "King's property".
Step 2: Key Formula or Approach:
The property states that .
Here, and . So, .
Step 3: Detailed Explanation:
Let the given integral be I.
Using the property , we replace x with .
We know that and .
Now, add equation (1) and equation (2):
Step 4: Final Answer:
The value of the integral is }.