Question

Evaluate 4 ſ (4x2 – 3x + 7) dx -1 is car S (4x2 – 3x + 7) dx = (Type an exact answer in simplified form.) -1

Answer 1

Since the function we want to integrate is a polynomial, we use the following general rule:

`\int ax^ndx=a\cdot(x^(n+1))/(n+1)`

Then, in this case we have:

`\begin{gathered} \int ^4_(-1)(4x^2-3x+7)dx=4\cdot(x^3)/(3)-3\cdot(x^2)/(2)+7x \\ =(4)/(3)x^3-(3)/(2)x^2+7x \end{gathered}`

Now we must evaluate F(b)-F(a). We have that a=-1 and b=4, then:

`\begin{gathered} F(x)=(4)/(3)x^2-(3)/(2)x^2+7x \\ a=-1 \\ b=4 \\ F(-1)=(4)/(3)\cdot(-1)^3-(3)/(2)\cdot(-1)^2+7(-1)=-(4)/(3)-(3)/(2)-7=-(59)/(6) \\ F(4)=(4)/(3)\cdot(4)^3-(3)/(2)\cdot(4)^2+7\cdot(4)=(256)/(3)-24+28=(256)/(3)+4 \\ =(268)/(3) \\ \Rightarrow F(b)-F(a)=(268)/(3)-(-(59)/(6))=(268)/(3)+(59)/(6)=(536+59)/(6)=(595)/(6) \end{gathered}`

Therefore, the integral of 4x^2-3x+7 evaluated from -1 to 4 is 595/6