Integral Calculus

Step 1: Express the right-hand side with a common denominator.
We are given: To combine the fractions on the right-hand side, we get a common denominator:

Step 2: Set the numerators equal.
Since the denominators on both sides are the same, we can set the numerators equal to each other:

Step 3: Expand both sides.
Expand the right-hand side: Now, combine like terms:

Step 4: Compare coefficients.
Equate the coefficients of and the constant terms on both sides:
- From the -terms:
- From the constant terms:

Step 5: Solve the system of equations.
We have the system:
Solve for and : - From , we get . - Substitute into : Now substitute into :

Step 6: Find .
Thus, , corresponding to option (B).

Step 1: Set up the partial fractions.
We are given the equation: To combine the fractions on the right-hand side, get a common denominator:

Step 2: Set the numerators equal.
Since the denominators are the same on both sides, we can set the numerators equal to each other:

Step 3: Expand both sides.
Expand the right-hand side: Now combine like terms:

Step 4: Compare coefficients.
Equate the coefficients of and the constant terms on both sides:
- From the -terms:
- From the constant terms:

Step 5: Solve the system of equations.
We have the system: Solve for and : - From , we get . - Substitute into : Now substitute into :

Step 6: Find .
Thus, , corresponding to option (B).

Step 1: Set up the partial fractions.
We are given the equation: To combine the fractions on the right-hand side, get a common denominator:

Step 2: Set the numerators equal.
Since the denominators are the same on both sides, we can set the numerators equal to each other:

Step 3: Expand both sides.
Expand the right-hand side: Now distribute :

Step 4: Combine like terms.
Combine the terms and constant terms:

Step 5: Compare coefficients.
Equate the coefficients of and the constant terms on both sides:
- From the -terms:
- From the constant terms:

Step 6: Conclusion.
Thus, , corresponding to option (D).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



First:
Then:


Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
Larger sides: . By Law of Sines:






By Law of Sines:

Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
Case 1: :
Case 2: :
Case 3: :
All cases yield .
Step 3: Final Answer

Step 1: Understanding the Concept:
Split the integral at to handle the modulus.

Step 2: Detailed Explanation:
For : The maximum occurs at the endpoints because is minimized at and maximized at endpoints.

Step 3: Final Answer:
Maximum value is , which corresponds to option (A).

Step 1: Understanding the Concept:
Use reduction formula for .
Step 2: Detailed Explanation:

Multiply by :
Thus
Step 3: Final Answer:
.

Step 1: Understanding the Concept:
Trapezoidal rule:
Step 2: Detailed Explanation:
, , ,
, , , ,

, , , ,
Approx =
Step 3: Final Answer:
42.

Step 1: Understanding the Concept:
Substitution method: let .
Step 2: Detailed Explanation:
Let , , ,

Integration by parts: Let , ,

So
Step 3: Final Answer:
.

Concept: Use substitution and standard identities.

Step 1: Use identity.


Step 2: Simplify integral.


Step 3: Apply integration formula.


Step 4: Compare with given form.

Concept: Simpson’s rule: Compute values: Substitute values and simplify:

Concept: Use substitution:

Step 1: Rewrite integral.


Step 2: Substitute.


Step 3: Use integration by parts.


Step 4: Back substitute.