Answer to Question #162930 in Calculus for Pia

Question #162930

∫x^2 sqrt(2-x)dx


1
Expert's answer
2021-03-02T00:55:31-0500

Let u=2xu=2-x , then du=dx-du=dx and x=2ux=2-u


Use the above substitution to evaluate the integral as,


x22xdx=(2u)2u(1)du\int x^2 \sqrt{2-x}dx=\int (2-u)^2 \sqrt{u}(-1)du


=(4+u24u)udu=-\int (4+u^2-4u) \sqrt{u}du


=(4u+u2u4uu)du=-\int (4\sqrt{u}+u^2\sqrt{u}-4u\sqrt{u})du


=[4(23u32)+27u724(25u52)]+C=-[4({\frac{2}{3}u^\frac{3}{2}})+{\frac{2}{7}u^\frac{7}{2}}-4({\frac{2}{5}u^\frac{5}{2}})]+C


=83u3227u72+85u52+C=-\frac{8}{3}u^\frac{3}{2}-\frac{2}{7}u^\frac{7}{2}+\frac{8}{5}u^\frac{5}{2}+C


=83(2x)3227(2x)72+85(2x)52+C=-\frac{8}{3}(2-x)^\frac{3}{2}-\frac{2}{7}(2-x)^\frac{7}{2}+\frac{8}{5}(2-x)^\frac{5}{2}+C



Therefore, the integral is x22xdx=83(2x)3227(2x)72+85(2x)52+C\int x^2 \sqrt{2-x}dx=-\frac{8}{3}(2-x)^\frac{3}{2}-\frac{2}{7}(2-x)^\frac{7}{2}+\frac{8}{5}(2-x)^\frac{5}{2}+C

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