Question

Find F as a function of x and evaluate it at x = 2, x = 5 and x = 8.

Answer 1

Given:

`F(x)=\int_2^x(t^3+6t-4)dt`

Find-:

`F(x),F(2),F(5),F(8)`

Sol:

`\begin{gathered} F(x)=\int_2^x(t^3+6t-4)dt \\ \\ \end{gathered}`

Use integration then:

`\begin{gathered} F(x)=\int_2^x(t^3+6t-4)dt \\ \\ F(x)=[(t^4)/(4)+(6t^2)/(2)-4t]_2^x^ \\ \\ \\ F(x)=(x^4)/(4)+3x^2-4x-(2^4)/(4)-3(2)^2+4(2) \\ \\ F(x)=(x)/(4)^4+3x^2-4x-8 \end{gathered}`

The function value at x = 2 is:

`\begin{gathered} F(x)=(x^4)/(4)+3x^2-4x-8 \\ \\ F(2)=(2^4)/(4)+3(2)^2-4(2)-8 \\ \\ F(2)=4+12-8-8 \\ \\ F(2)=16-16 \\ \\ F(2)=0 \end{gathered}`

The function value at x = 5

`\begin{gathered} F(x)=(x^4)/(4)+3x^2-4x-8 \\ \\ F(5)=(5^4)/(4)+3(5)^2-4(5)-8 \\ \\ F(5)=156.25+75-20-8 \\ \\ F(5)=203.25 \end{gathered}`

Function value at x = 8

`\begin{gathered} F(x)=(x^4)/(4)+3x^2-4x-8 \\ \\ F(8)=(8^4)/(4)+3(8)^2-4(8)-8 \\ \\ F(8)=1024+192-32-8 \\ \\ F(8)=1216-40 \\ \\ F(8)=1176 \end{gathered}`