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This is the opposite of the derivative - and it's an integral part of Calculus. It's uses range from basic integrals to differential equations, with applications in Physics, Chemistry, and Economics.

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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