Evaluate this Integral
x(x-1)^4 with upper limit 2 and lower limit 1.
Use the substitution u=x-1
Answer should be 11/30, but I get no where near that, I think I may be integrating it wrong
Thanks in advance.
Hello, Hoopy!
We could find your error if you would show your work . . .
Evaluate: .$\displaystyle \int^2_1x(x-1)^4\,dx$
Let: $\displaystyle u = x-1\quad\Rightarrow\quad x = u+1\quad\Rightarrow\quad dx = du$
Substitute: .$\displaystyle \int^1_0(u+1)u^4\,du \;=\;\int^1_0\left(u^5 + u^4\right)\,du\;= \;\frac{1}{6}u^6 + \frac{1}{5}u^5\,\bigg]^1_0 \;=\;\frac{11}{30}$
$\displaystyle \int^2_1 x(x-1)^4 \, \mathrm{d}x$
$\displaystyle u=x-1 \therefore \frac{\mathrm{d}u}{\mathrm{d}x} = 1 \implies {\mathrm{d}u}={\mathrm{d}x}$
Replace $\displaystyle x(x-1)^4$ with $\displaystyle u$ and $\displaystyle x$ with $\displaystyle u + 1$.
$\displaystyle \therefore limits: x=2 \implies u= x-1 = 2-1 =1$
$\displaystyle \therefore limits: x=1 \implies u= x-1 = 1-1 =0$
$\displaystyle \therefore \int^1_0 (u+1)(u)^4 \, \mathrm{d}u$
$\displaystyle = \int^1_0 (u^5 + u^4) \, \mathrm{d}u $
$\displaystyle = [\frac{u^6}{6} + \frac{u^5}{5}]^1_0$
$\displaystyle =[\frac{11}{30}] - [0] = \frac{11}{30}$