Math  /  Calculus

Questionb) [6 pts] Let f be a continuous function such that: arccos(x8)arctan(x4)+x2=0xf(t)dt.\arccos \left(\sqrt{\frac{x}{8}}\right) \arctan \left(\frac{x}{4}\right)+x^{2}=\int_{0}^{\sqrt{x}} f(t) d t .
Find f(2)f(2).

Studdy Solution
Evaluate f(2) f(2) :
Since f(x) f(\sqrt{x}) is expressed in terms of x x , substitute x=4 x = 4 to find f(2) f(2) :
f(2)=24(uv+uv+2×4) f(2) = 2\sqrt{4} \left( u'v + uv' + 2 \times 4 \right)
Calculate each term and substitute:
f(2)=4(uv+uv+8) f(2) = 4 \left( u'v + uv' + 8 \right)
Substitute the derivatives and simplify to find the value of f(2) f(2) .
The value of f(2) f(2) is:
16 \boxed{16}

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