91Ó°ÊÓ

Assuming that \(\int g(x) d x=G(x)+C,\) where \(G(4)=9\) \(G(6)=4,\) and \(G(9)=6,\) evaluate the definite integral: (a) \(\int_{6}^{4} g(x) d x\) (b) \(\int_{6}^{9} 7 g(x) d x\) (c) \(\int_{4}^{9}(g(x)+3) d x\)

For Problems \(97-99,\) let \(\int g(x) d x=G(x)+C .\) Which of (I)-(III), if any, is equal to the given integral? \(\int g(2 x) d x\) I. \(0.5 G(0.5 x)+C\) II. \(0.5 G(2 x)+C\) III. \(2 G(0.5 x)+C\)

For Problems \(97-99,\) let \(\int g(x) d x=G(x)+C .\) Which of (I)-(III), if any, is equal to the given integral? \(\int \cos (G(x)) g(x) d x\) I. \(\sin (G(x)) g(x)+C\) II. \(\sin (G(x)) G(x)+C\) III. \(\sin (G(x))+C\)

In Problems \(102-103,\) give an example of: Two different functions \(F(x)\) and \(G(x)\) that have the same derivative.

In Problems \(102-103,\) give an example of: A function \(f(x)\) whose antiderivative \(F(x)\) has a graph which is a line with negative slope.

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. An antiderivative of \(3 \sqrt{x+1}\) is \(2(x+1)^{3 / 2}\).

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. An antiderivative of \(3 x^{2}\) is \(x^{3}+\pi\).

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. An antiderivative of \(1 / x\) is \(\ln |x|+\ln 2\).

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. If \(F(x)\) is an antiderivative of \(f(x)\) and \(G(x)=F(x)+2\) then \(G(x)\) is an antiderivative of \(f(x)\).

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. If \(F(x)\) and \(G(x)\) are two antiderivatives of \(f(x)\) for \(-\infty < x < \infty\) and \(F(5) > G(5),\) then \(F(10) > G(10)\).