Integral

Step 1: Split the integral into two parts based on the absolute value function: Step 2: Simplify each integral: Step 3: Now, compute each integral: Step 4: Adding these results gives:

Step 1: Simplify the integrand using the trigonometric identity for : Step 2: This integral can now be solved directly:

Step 1: The greatest integer function gives the largest integer less than or equal to . The value of changes at each integer value of . Step 2: Break the integral into parts where remains constant: Step 3: Now, calculate each integral: Step 4: Adding these values gives:

We can use a trigonometric identity to simplify the integrand. Recall that: Thus, the integral becomes: The integral of is , so we have: Alternatively, a direct approach would lead to:

Concept: Recognize derivative form.

Step 1: Differentiate .


Step 2: Match integrand.

Concept: Recognize derivative inside cosine.

Step 1: Let .

Step 2: Then .

Step 3: Substitute.

Concept: Let

Step 1: Substitute.


Step 2: Integrate.

Concept: Use substitution.

Step 1: Let .


Step 2: Substitute.


Step 3: Integrate.

Concept: Split into:

Step 1: Split integral.


Step 2: Solve.

Step 1: Understanding the Concept:
Use substitution . Then .

Step 2: Detailed Explanation:


Step 3: Final Answer:
.

Step 1: Understanding the Concept:
Use the identity .

Step 2: Detailed Explanation:



Step 3: Final Answer:
.

Step 1: Concept:
• Use substitution:

Step 2: Detailed Explanation:
• Given integral:
• Rewrite:
• Let:
• Substitute:
• Split using partial fractions:
• Integrate:
• Simplify:

Step 3: Final Answer:
•

Step 1: Concept:
• Multiply numerator and denominator by .

Step 2: Detailed Explanation:
• Given integral:
• Multiply numerator and denominator by :
• Let:
• Substitute:
• Integrate:

Step 3: Final Answer:
•

Step 1: Concept:
• Use substitution:
• Then:

Step 2: Detailed Explanation:
• Substitute into the integral:
• This becomes:
• Integrate:
• Substitute back:

Step 3: Final Answer:
•

We can use substitution to solve this integral. Let: This transforms the integral into: Substituting back for :

The function represents the greatest integer less than or equal to . Since the integral is over a symmetric interval, and the floor function is an odd function, the areas on either side of the origin cancel out. Thus, the integral evaluates to 0.

Concept:
• Use substitution:

Step 1: Substitute


Step 2: Substitute in integral


Step 3: Integrate


Step 4: Back substitute
Final Conclusion:
Option (B)

Concept:
• Recognize derivative form of

Step 1: Let function be


Step 2: Differentiate


Step 3: Match integrand
Final Conclusion:

Concept:
• Use identity:

Step 1: Apply identity


Step 2: Integrate
Final Conclusion:

Concept:
• Use polynomial division before integrating

Step 1: Divide by


Step 2: Integrate term-wise
Final Conclusion:
Option (C)

Concept:
• Use substitution:

Step 1: Let


Step 2: Rewrite denominator
Using identity simplification gives:

Step 3: Substitute


Step 4: Use standard form
Here

Step 5: Back substitute
Final Conclusion:
Option (A)

Step 1: Understanding the Concept:
Standard trigonometric integrals often require converting powers of and into and forms, as the latter have direct antiderivatives.

Step 2: Key Formula or Approach:
1.
2.

Step 3: Detailed Explanation:
1. Substitute the identities into the integral:
2. Integrate each term separately. Recall that and :
3. Let's re-verify the sign in the original expression: . The expression above gives , which matches option (e). However, standard keys for this specific problem often simplify using via different identities. Let's stick to the direct derivation: .

Step 4: Final Answer
The integral evaluates to .

Step 1: Understanding the Concept:
We are given an indefinite integral that involves a composite function under a square root and an algebraic fraction.
The presence of inside the root and outside strongly suggests the method of substitution, as the derivative of yields a term proportional to .

Step 2: Key Formula or Approach:
Use the integration by substitution method: Let , then .

Step 3: Detailed Explanation:
Let .
Differentiating both sides with respect to , we get:
This implies that .
We can rearrange this to match the term in our integral:
Now, substitute and back into the original integral:
Integrate using the power rule :
Finally, substitute back the original expression for :


Step 4: Final Answer:
The calculated integral matches option (B).