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Math Help - Can someone please explain the steps in this integral problem?

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by operaphantom2003 View Post
    ∫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.
    \int x(x^2+1)^5 \, dx

    if u = x^2+5, then du = 2x \, dx

    multiply the integral by \frac{1}{2} \cdot 2 ... the \frac{1}{2} outside the integral, the " 2" inside to give you du

    \frac{1}{2} \int 2x(x^2+1)^5 \, dx

    now substitute ...

    \frac{1}{2} \int u^5 \, du

    antiderivative is ...

    \frac{1}{2} \cdot \frac{u^6}{6} + C

    multiply the fractions ... see where the \frac{u}{12} comes from?
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