Calculus

Antiderivatives

Integration of Algebraic Functions

         

Which function f(x)f(x) satisfies f(x)=6x28x+9,f'(x)=6x^2-8x+9, f(1)=12?f(1)=12?

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

Suppose there are functions F(x)F(x) and f(x)f(x) such that f(x)=6x2+6xf(x) = 6x^2 + 6x represents the slope of the tangent line to F(x)F(x) for all xx. If the graph of y=F(x)y = F(x) passes through the point (1, 8), then what is the value of F(3)F(3)?

If constants aa, bb, and cc satisfy ddx(ax2+7x+4) dx=5x2+bx+c,\frac{d}{dx} \int (ax^2+7x+4)\ dx = 5x^2+bx+c, what is a+b+ca+b+c?

If f(x)=(1+2x+3x2++8x7)dxf(x)=\int \left( 1+2x+3x^2+\cdots + 8x^{7} \right)dx and f(0)=8,f(0)=8, what is the value of f(2)?f(2)?

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