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Calculus

# 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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