An Equation of Integrals

Calculus Level 3

If f f is a continuous function such that the following holds for all x x ∫ 0 x f ( t ) d t = x sin ⁡ ( x ) + ∫ 0 x f ( t ) 1 + t 2 d t , \int_0^x f(t)\,dt = x\sin(x)+\int_0^x \frac{f(t)}{1+t^2}\,dt, then evaluate f ( π 2 ) f(\frac{\pi}{2}) .


The answer is 1.405.

Vincent Moroney by Vincent Moroney , 22, Canada

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1 solution

Vincent Moroney
Jun 25, 2018

Differentiate both sides of the equation to give f ( x ) = sin ⁡ ( x ) + x cos ⁡ ( x ) + f ( x ) 1 + x 2 ⇒ f ( x ) ( x 2 1 + x 2 ) = sin ⁡ ( x ) + x cos ⁡ ( x ) f(x) = \sin(x)+x\cos(x) + \frac{f(x)}{1+x^2} \Rightarrow f(x)\Big(\frac{x^2}{1+x^2}\Big) = \sin(x) + x\cos(x) f ( x ) = 1 + x 2 x 2 ( sin ⁡ ( x ) + x cos ⁡ ( x ) ) ⇒ f ( π 2 ) = 1.405 . f(x) = \frac{1+x^2}{x^2}(\sin(x) + x\cos(x)) \Rightarrow f(\frac{\pi}{2}) = \boxed{1.405}.

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