Question

Use Part Il of the Fundamental Theorem of Calculus to evaluate the definite integral

Answer 1

`4\ln (2)+(49)/(3)\approx19.1059`

Given:

`\int ^(-1)_(-2)(7x^5-4x^2)/(x^3)dx`

Simplify:

`\int (7x^3-4)/(x)dx`

Expand:

`\int (7x^2-(4)/(x))dx`

Apply linearity:

`7\int x^2dx-4\int (1)/(x)dx`

Apply power rule and the standard integral ln(x)

`7((x^3)/(3))-4\ln (x)`

Now, applying the Fundamental Theorem of Calculus Part 2

`\int ^(-1)_(-2)(7x^5-4x^2)/(x^3)dx=(7(((-1)^3)/(3))-4\ln (-1))-(7(((-2)^3)/(3))-4\ln (-2))`
`=4\ln (2)+(49)/(3)`

Or approximately

`\approx19.1059`