Hello, powp!
I have the following problem. Can anybody help?
. . . \(\displaystyle \L\int \frac{x^2}{\sqrt{1-x}}\,dx\)
My first and only thought was to subsitute 1 - x with U,
. . but that gave me an answer not ever close.
pka had the best suggestion, but your method should have worked . . .
I bet your answer was correct, but you couldn't make it match
their answer.
\(\displaystyle \text{Let }\L u\,=\,1-x\;\;\Rightarrow\;\;x\,=\,1-u\;\;\Rightarrow\;\;dx\,=\,-du\)
\(\displaystyle \text{Substitute: }\L\int\frac{(1-u)^2}{u^{\frac{1}{2}}}(-du)\;=\;-\int\frac{1\,-\,2u\,+\,u^2}{u^{\frac{1}{2}}}\,du\)
. . \(\displaystyle \L=\;-\int\left(u^{-\frac{1}{2}}\,-\,2u^{\frac{1}{2}}\,+\,u^{\frac{3}{2}}\right)\,du\;=\;-\left(2u^{\frac{1}{2}}\,-\,\frac{2}{3}u^{\frac{3}{2}}\,+\,\frac{2}{5}u^{\frac{5}{2}}\right)\,+\,C\)
\(\displaystyle \text{Factor out }\frac{2}{15}:\;\;\L-\frac{2}{15}\left(15u^{\frac{1}{2}}\,-\,10u^{\frac{3}{2}}\,+\,3u^{\frac{5}{2}}\right)\,+\,C\)
\(\displaystyle \text{Factor out }u^{\frac{1}{2}}:\;\;\L-\frac{2}{15}u^{\frac{1}{2}}\left(15\,-\,10u\,+\,3u^2\right)\,+\,C\)
\(\displaystyle \text{Back-substitute: }\;\L-\frac{2}{15}(1-x)^{\frac{1}{2}}\left[15\,-\,10(1\,-\,x)\,+\,3(1\,-\,x)^2\right]\,+\,C\)
\(\displaystyle \text{which simplifies to: }\;\L-\frac{2}{15}\,\sqrt{1\,-\,x}\,(8\,+\,4x\,+\,3x^2)\,+\,C\)
