Find f(4) if

Find f(4) if

find f(4) if
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Re: Find f(4) if

I would start by differentiating both sides of the equation with respect to x:
$$2xf(x^2)= cos(\pi x)- \pi x sin(\pi x)$$.

Now set x= 2 and solve for f(4).

(Differentiate the integral using the "Fundamental Theorem of Calculus", $$\frac{d}{du}\int_0^u f(t)dt= f(u)$$ together with the substitution $$u= x^2$$ and the chain rule.)
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Re: Find f(4) if

"Leibniz's rule" is a generalization of the "Fundamental Theorem of Calculus":

$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x, t)dt= \frac{d\beta(x)}{dx}f(x,\beta(x)- \frac{d\alpha(x)}{dx}f(x, \alpha(x))+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f(x,t)}{\partial x} dt$$.

In this problem, $$\alpha(x)= 0$$, $$/beta(x)= x^2$$, f is a function of t only so that reduces to $$\frac{d}{dx}\int_0^{x^2} f(t)dt= 2x f(x^2)= x cos(\pi x)$$.
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