Thread: Can someone please explain the steps in this integral problem?

∫x(x^2+1)^5 dx from 0 to 2

Part I know:
substitution is u=x^2+1, du=2x dx, x=0 u=1, x=2 u=5

Part from solutions manual:
(1/2) ∫ (x^2+1)^5 (2x dx)
(1/2) ∫ u^5 du from x=1 to 5
[u/12] from 1 to 5
(5^6-1^6)/12
15624/12
answer = 1302

Now my questions:
Where did the 1/2 in front of the integral come from?
Where did the u/12 come from?
Is there a trick to this that I am just not seeing? -- Calc 2 is kicking my butt so far.

Originally Posted by operaphantom2003

∫x(x^2+1)^5 dx from 0 to 2

Part I know:
substitution is u=x^2+1, du=2x dx, x=0 u=1, x=2 u=5

Part from solutions manual:
(1/2) ∫ (x^2+1)^5 (2x dx)
(1/2) ∫ u^5 du from x=1 to 5
[u/12] from 1 to 5
(5^6-1^6)/12
15624/12
answer = 1302

Now my questions:
Where did the 1/2 in front of the integral come from?
Where did the u/12 come from?
Is there a trick to this that I am just not seeing? -- Calc 2 is kicking my butt so far.

if , then

multiply the integral by ... the outside the integral, the " " inside to give you



now substitute ...



antiderivative is ...



multiply the fractions ... see where the comes from?