Calculus

Antiderivatives

Antiderivative and Indefinite Integration

         

If two polynomial functions f(x)f(x) and g(x)g(x) satisfy ddx{f(x)+g(x)}=7,ddx{f(x)g(x)}=12x1,f(0)=0,g(0)=1,\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)?f(1)-g(1)?

If f(x)={ddx(x35x2+18x)}dx and f(1)=2, 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)?f(3)?

If a,b,a ,b, and cc are constants satisfying (12x3+ax9)dx=bx4+3x2+cx+C,\int \left( 12x^3 +ax-9\right)dx=bx^4+3x^2+cx+C, what is the value of a+b+c?a+b+c?

Details and assumptions

CC is the constant of integration.

If a,ba, b and cc are constants and (18x2+ax+1)dx=bx3+8x2+cx+C,\int (18x^2+ax+1)dx=bx^3+8x^2+cx+C, where CC is the constant of integration, what is a+b+c?a+b+c?

If f(x)f(x) satisfies (2x+1)f(x)dx=x42x32x2+C,\int (2x+1)f(x)dx = x^4-2x^3-2x^2+C, what is the value of f(5)?f(5)?

Details and assumptions

CC is the constant of integration.

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