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Antiderivatives

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.

Antiderivative and Indefinite Integration

         

If two polynomial functions \(f(x)\) and \(g(x)\) satisfy \[\frac{d}{dx}\{f(x)+g(x)\}=7, \frac{d}{dx}\{f(x)g(x)\}=12x-1, f(0)=0, g(0)=-1,\] what is \(f(1)-g(1)?\)

If \[ f(x)=\int \left\{\frac{d}{dx} \left(x^3-5x^2+18x\right)\right\} dx \text{ and } f(1)=2,\] what is \(f(3)?\)

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 \(a, b\) and \(c\) are constants and \[\int (18x^2+ax+1)dx=bx^3+8x^2+cx+C,\] where \(C\) is the constant of integration, what is \(a+b+c?\)

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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