Question
Mathematics Question on limits and derivatives
Let I(x)=∫sin2x(1−cotx)26dx. If I(0)=3, then I(12π) is equal to:
A
3
B
33
C
63
D
23
Answer
33
Explanation
Solution
Given : I(x)=∫sin2x(1−cotx)26dx.
Step 1: Substitution Let: t=1−cotx,csc2xdx=dt. The integral becomes: I=∫t26dt=−t6+c=−1−cotx6+c.
Step 2: Using I(0)=3: At x=0, cot(0)=∞. Substituting: I(0)=3=−1−cot(0)6+c⟹c=3. Thus, the expression for I(x) becomes: I(x)=3−1−cotx6.
Step 3 : Evaluate I(12π): At x=12π: cot(12π)=2+3. Substitute into I(x): I(12π)=3−1−(2+3)6. Simplify: I(12π)=3+2+3−16=3+1+36.
Step 4 : Rationalize the denominator: 1+36=(1+3)(1−3)6(1−3)=1−36(1−3)=26(3−1)=3(3−1). Substitute back: I(12π)=3+33−3=33.
Final Answer: 33.