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.

If \(a ,b,\) and \(c\) are constants satisfying \[\int \left( 12x^3 +ax-9\right)dx=bx^4+3x^2+cx+C,\] what is the value of \(a+b+c?\)

**Details and assumptions**

\(C\) is the constant of integration.

If \(f(x)\) satisfies \[\int (2x+1)f(x)dx = x^4-2x^3-2x^2+C,\] what is the value of \(f(5)?\)

**Details and assumptions**

\(C\) is the constant of integration.

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