Definite Integral

Step 1: Integral of the given form
The integral is a standard result:
Step 2: Substitute limits
Here, , so:
Step 3: Evaluate the bounds
Since and , we get:
Step 4: Verify the options
The correct value is , which corresponds to option (C).

Step 1: Apply symmetry property of definite integrals
Let: Using the property , we get:
Step 2: Combine integrals
Adding the two forms of :
Step 3: Simplify the integrand
Rewrite : and simplify:
Step 4: Integrate and simplify
The integral evaluates to: Substituting limits:
Conclusion: The integral evaluates to .

We are given the curve and we are to find the area enclosed between and , bounded below by the -axis ( ). Since the curve lies below the -axis between and , the area is given by: Evaluating the integral: % Final Answer Final Answer: The required area is square units. % Rough Sketch \begin{center} \includegraphics[width=0.6\textwidth]{ig4.png} \end{center}