Trigonometry

Step 1: Let . Substitute this into the given inequality. The inequality becomes: Step 2: Find the roots of the corresponding quadratic equation . We can solve this by factoring or using the quadratic formula. Let's factor the quadratic expression: We look for two numbers that multiply to and add up to . These numbers are and . The roots are and . Step 3: Determine the interval where the quadratic expression is less than 0. Since the coefficient of (which is 4) is positive, the parabola opens upwards. Therefore, the expression is negative between its roots. Step 4: Substitute back . This means that lies in the open interval . Step 5: Compare the result with the given options. The interval matches option (B). Option (A) is invalid as . Option (C) contains the correct interval but is wider.

We are given the integral: To solve this, we use the substitution method. Let: Substituting this into the integral, we get: We know that: Therefore, the integral becomes: Thus, the result matches when the value of . The correct answer is .

Step 1: Use the identity.

Step 2: Substitute .
Using standard trigonometric values,
Step 3: Perform subtraction.

Step 4: Simplify using identity.
Alternatively,

Step 1: Use known trigonometric identity.

Step 2: Substitute value of .

Step 3: Apply product identity.

Step 4: Final calculation.