Question

Find the antiderivative of ..?

`(6x^5-7x^3+12x)/(2x)`

Answer 1

`int(6x^5-7x^3+12x)/(2x)dx=(6x^5)/10-(7x^3)/6+6x+"C"`

Explanation:

Given: `int(6x^5-7x^3+12x)/(2x)dx`

Take out the constant `1/2`

`1/2*int(6x^5-7x^3+12x)/xdx`

We can separate each term and integrate each term individually so we can rewrite the integral as:

`1/2[int(6x^5)/xdx-int(7x^3)/xdx+int(12x)/xdx]`

But before we integrate, simplify!

`1/2[int6x^4dx-int7x^2dx+int12dx]`

Now we can integrate each term:

`int6x^4dx=>intcx^adx=(cx^(a+1))/(a+1)=>(6x^(4+1))/(4+1)=(6x^5)/5`

`-int7x^2dx=>intcx^adx=(cx^(a+1))/(a+1)=>(7x^(2+1))/(2+1)=-(7x^3)/3`

`int12dx=>intcdx=cx=>=12x`

Putting it all together we get

`=1/2[(6x^5)/5-(7x^3)/3+6x]`

`=(6x^5)/10-(7x^3)/6+6x+"C"larr` Don't Forget the Constant!