AP EAMCET · Maths · Definite Integration
\(\int_{-\pi}^\pi \frac{\cos ^{2022} x}{1+(2022)^x} d x=\)
- A \(\frac{(2022) !}{2^{2022}((1011) !)^2} \pi\)
- B \(\left({ }^{2022} C_{1011}\right) \pi\)
- C \(\left({ }^{2022} C_{1011}\right) \frac{\pi}{2^{1011}}\)
- D \(\frac{(2022) !}{(1011) ! 2^{2022}} \pi\)
Answer & Solution
Correct Answer
(A) \(\frac{(2022) !}{2^{2022}((1011) !)^2} \pi\)
Step-by-step Solution
Detailed explanation
\(I=\int_{-\pi}^\pi \frac{\cos ^{2022}}{1+(2022)^x} d x\) ...(i) \(\Rightarrow I=\int_{-\pi}^\pi \frac{\cos ^{2022} x \cdot\left(2022^x\right.}{\left(202 Z^x+1\right.} d x\) ...(ii) On adding Eqs. (i) and (ii), \(2 I=\int_{-\pi}^\pi \cos ^{2022} x d x\)…
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