29Thats direct application of Leibniz formula
\int_{sinx}^{1}{t^{2}f(t)dt} = 1- sinx
Differentiating both sides wrt to x we will get
-sin^{2} x f(sinx)cosx= - cosx
sinx =1/√3
just substitute the value u will get the answer
1. \int_{sinx}^{1}{t^{2}f(t)dt}=1-sinx,0\leq x\leq \frac{\pi }{2}, then f\left(\frac{1}{\sqrt{3}} \right)=
ans ---------> 3
2. Function f is continuous for all x,( and not everywhere 0), such that \left\{f(x) \right\}^{2} = \int_{0}^{x}{\frac{f(t)sint}{2+cost}}dt,then prove that f(x) = 12ln32+cosx
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2 Answers
2929Use Leibniz formula for the second one also...
On differentiating both sides wrt x u will get
2 f(x)f'(x) = \frac{f(x)sinx}{2+cosx} \Rightarrow 2 f'(x) = \frac{sinx}{2+cosx}
Now integrate it using 2+cosx = t..
