I want to find the integral of $x (1-x)^8$. How do I go about this? For example, which rule do I use from http://integral-table.com ? Thanks!
What is the integral of $x(1-x)^8$?
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1If you really want to use that table, then you could try (7). Note that $(1 - x)^8 = (x - 1)^8$, as $(-1)^8 = 1$. – 2011-11-12
4 Answers
$\begin{align*} \int x(1-x)^8\,dx &=\int x(x-1)^8\,dx\\ &=\int (x-1+1)(x-1)^8\,dx\\ &=\int ((x-1)^9+(x-1)^8)\,dx\\ &=\frac1{10}(x-1)^{10}+\frac19(x-1)^9+C\quad (C: \text{constant}). \end{align*}$
Note that in the last line I used $\int (x+a)^n\,dx=\frac1{n+1}(x+a)^{n+1}+C,\quad \text{when}\ n\ne -1.$
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0Ultimately, @pharmine rearranged the expression from step 1 to step 3 in order to simplify the problem using substitution techniques. – 2011-11-13
Nobody seems to have mentioned what I'd have thought was an obvious point: It's simpler if the thing raised to the 8th power is just a variable rather than $1$ minus a variable, so let $ \begin{align} u & = 1-x, \\ \\ du & = - dx, \\ \\ x & = 1-u. \end{align} $ Then $ \int x (1-x)^8\;dx = \int (1-u)u^8\;(-du) = \int u^9-u^8\;du. $ It's easy to antidifferentiate that, and then put $1-x$ wherever $u$ appears.
(After that, the 10th and 9th powers of $1-x$ can be expanded if desired.)
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1The remark is also apparent in a previous answer. – 2011-11-12
Method $1$: Use the binomial expansion of $(1-x)^{8}$ and then integrate term by term.
Method $2$: Use integration by parts, by taking $u=x$ and $dv = (1-x)^{8} \ dx$.
$x(1-x)^8 = x-8 x^2+28 x^3-56 x^4+70 x^5-56 x^6+28 x^7-8 x^8+x^9$
so
$\begin{array}{cl} && \int x(1-x)^8 \,dx \\ &=& \int x-8 x^2+28 x^3-56 x^4+70 x^5-56 x^6+28 x^7-8 x^8+x^9 \,dx \\\\ &=& \int x \,dx - 8 \int x^2 \,dx +28 \int x^3 \,dx - 56 \int x^4 \,dx + 70 \int x^5 \,dx - 56 \int x^6 \,dx + 28 \int x^7 \,dx - 8 \int x^8 \,dx + \int x^9 \,dx \\ &=& \tfrac{1}{2} x^2 - \tfrac{8}{3} x^3 + \tfrac{28}{4} x^4 - \tfrac{56}{5}x^5 + \tfrac{70}{6} x^6 - \tfrac{56}{7} x^7 + \tfrac{28}{8} x^8 - 8 \tfrac{1}{9} x^9 + \tfrac{1}{10} x^{10} + C \end{array}$
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2No cleverness, but I'm afraid this margin is too small all the same. – 2011-11-12