Question

How do you evaluate the definite integral `int (x^3+x-6)dx` from [2,4]?

Answer 1

I found: `int_2^4(x^3+x-6)dx=54`

Explanation:

First we can evaluate the integral and then substitute the extremes of integration and subtract the values obtained (Fundamental Theorem of Calculus):
`int(x^3+x-6)dx=` separate:
`=intx^3dx+intxdx-int6dx=`
integgrate each one:
`=x^4/4+x^2/2-6x`
For:

`x=4`
we get:`=4^4/4+4^2/2-6*4=64-8-24=48`

`x=2`
we get:`=2^4/4+2^2/2-6*2=4+2-12=-6`

Finally, subtract the two values:
`48-(-6)=54`

So:
`int_2^4(x^3+x-6)dx=54`

This value represents the Area below the curve defined by your function between `x=2` and `x=4`: