Question

How do you evaluate the definite integral `int (xsqrtx)dx` from [4,9]?

Answer 1

The answer is `=422/5`

Explanation:

To evaluate this integral, we use

`intx^ndx=x^(n+1)/(n+1)+C`

`xsqrtx=x*x^(1/2)=x^(3/2)`

`int_4^9(xsqrtx)dx=int_4^9x^(3/2)dx= [x^(5/2)/(5/2) ]_4^9`

`= [ 2/5x^(5/2) ]_4^9`

`=2/5(3^5-2^5)`

`=(486-64)/5`

`=422/5`

Answer 2

`422/5`

Explanation:

Express `xsqrtx=x xxx^(1/2)=x^(3/2)`

`rArrint_4^9x^(3/2)dx`

integrate using the `color(blue)"power rule for integration"`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(intax^ndx=a/(n+1)x^(n+1))color(white)(2/2)|)))`

`rArrint_4^9x^(3/2)dx=[1/(5/2)x^(5/2)]_4^9=[2/5x^(5/2)]_4^9`

`=2/5(9^(5/2)-4^(5/2))`

`=2/5(243-32)=422/5`