WBJEE · Maths · Definite Integration
Let \(f\) be a non-negative function defined on \(\left[0, \frac{\pi}{2}\right]\). If \(\int_0^x\left(f^{\prime}(t)-\sin 2 t\right) d t=\int_x^0 f(t) \tan t d t, f(0)=1\), then \(\int_0^{\frac{\pi}{2}} f(x) d x\) is
- A 3
- B \(3-\frac{\pi}{2}\)
- C \(3+\frac{\pi}{2}\)
- D \(\frac{\pi}{2}\)
Answer & Solution
Correct Answer
(B) \(3-\frac{\pi}{2}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} \text {Hint } & f^{\prime}(x)-\sin 2 x=-f(x) \tan x \\ & \Rightarrow f^{\prime}(x)+\tan x f(x)=\sin 2 x \\ & \Rightarrow \frac{d f(x)}{d x}+\tan x . f(x)=\sin 2 x \end{aligned}\)…
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