JEE Mains · Maths · STD 12 - 7.2 definite integral
If for a continuous function \(f(x),\) \(\int\limits_{ - \pi }^t {(f(x) + x\,\,dx)} = {\pi ^2} - {t^2},\) for all \(t\, \ge - \pi ,\) then \(f\left( { - \frac{\pi }{3}} \right)\) is equal to
- A \(\pi \)
- B \(\frac {\pi }{2}\)
- C \(\frac {\pi }{3}\)
- D \(\frac {\pi }{6}\)
Answer & Solution
Correct Answer
(A) \(\pi \)
Step-by-step Solution
Detailed explanation
Let \(\int_{-\pi}^{t}(f(x)+x) d x=\pi^{2}-t^{2}\) \(\Rightarrow \int_{-\pi}^{t} f(x) d x+\int_{-\pi}^{t} x d x=\pi^{2}-t^{2}\) \(\Rightarrow \int_{-\pi}^{t} f(x) d x+\left(\frac{t^{2}}{2}-\frac{\pi^{2}}{2}\right)=\pi^{2}-t^{2}\)…
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