Can someone check my work and answer?
Evaluate the integral from -1 to 0 of (4x^6+2x)^3(12x^5+1)dx
My work:
let u=4x^6+2x
dx=du/24x^5+2
now we have the integral from -1 to 0 of u^3(12x^5+1)(du/24x^5+2)
Simplifies to
the integral from -1 to 0 of u^3(1/2+1/2)
simplifies to
the integral from -1 to 0 of u^3
Apply the integral power rule and you'll have
the integral from -1 to 0 of u^4/4
Put the limits into terms of u by pluggin each into the substituted equation. I got 0 for the upper limit and 2 for the lower.
Apply the fundamental theorem of calculus and you'll get 0-4=-4.
Every calculator I've put this in has told me it's -2 though. Can someone show me if this is right or where I went wrong?
OK. Let's see here...
∫(4x^6+2x)^3(12x^5+1)dx
let
u = 4x^6+2x
du = 24x^5+2 = 2(12x^5+1)
so, you now have
∫1/2 u^3 du
= 1/8 u^4 [-1,0]
= 1/8 (4x^6+2x)^4 [-1,0]
= 1/8 (0-(4-2)^4)
= 1/8 (-16)
= -2
where did you get u^3(1/2+1/2) ? There's only the one term.
Another way is to change the limits of integration. Since
u = 4x^6+2x,
when x=-1, u=2
when x=0, u=0
Then we have
∫[-1,0] (4x^6+2x)^3(12x^5+1)dx
= ∫[2,0] 1/2 u^3 du
= 1/8 u^4 [2,0]
= 1/8 (0-16)
= -2
Oh I see. I was dividing 12 by 24 and 1 by 2 to get the 1/2 + 1/2. I didn't even think of factoring out the 2. Thank you!
To evaluate the integral from -1 to 0 of (4x^6+2x)^3(12x^5+1)dx, you made the correct substitution, u = 4x^6+2x.
However, when you substitute dx = du/(24x^5+2), you should be careful with the denominator. It should be 24u^5 + 2, not 24x^5 + 2. This mistake carries through the rest of your work and is the reason for the discrepancy in your answer.
So now, we have the integral from -1 to 0 of u^3(12x^5+1)(du/(24u^5+2)). Simplify this to the integral from -1 to 0 of (u^3/2u^5+1/2)du.
You correctly simplified to the integral from -1 to 0 of u^3.
Now, we can apply the power rule for integration. The integral of u^n du is (u^(n+1))/(n+1), so the integral from -1 to 0 of u^3 is [(u^4)/4] evaluated from -1 to 0.
Plugging in the limits gives us [(0^4)/4] - [(-1^4)/4] = 0 - (-1/4) = 1/4.
Therefore, the correct evaluation of the integral is 1/4, not -4 or -2.