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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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