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Integration of Algebraic Functions


Which function \(f(x)\) satisfies \[f'(x)=6x^2-8x+9,\] \[f(1)=12?\]

Suppose \(f(x) = 10x + 4\) is a function such that \(F'(x) = f(x)\). If the graph of the function \(y = F(x)\) passes through the origin, then what is the value of \(F(3)\)?

Suppose there are functions \(F(x)\) and \(f(x)\) such that \(f(x) = 6x^2 + 6x\) represents the slope of the tangent line to \(F(x)\) for all \(x\). If the graph of \(y = F(x)\) passes through the point (1, 8), then what is the value of \(F(3)\)?

If constants \(a\), \(b\), and \(c\) satisfy \[\frac{d}{dx} \int (ax^2+7x+4)\ dx = 5x^2+bx+c,\] what is \(a+b+c\)?

If \(f(x)=\int \left( 1+2x+3x^2+\cdots + 8x^{7} \right)dx\) and \(f(0)=8,\) what is the value of \(f(2)?\)


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