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xz_007 [3.2K]
2 years ago
7

Evaluate the integral. pls help me see the steps. thank you​

Mathematics
1 answer:
Alona [7]2 years ago
8 0

Answer:

-\frac{2}{15}(1-x)^\frac{3}{2}(3x+2)+C

Step-by-step explanation:

Use the method of integration by parts. The method is based in the following rule of derivation of two functions f and g :

(fg)' = f'g+fg'\implies fg'=(fg)'-f'g\implies \\\int fg' dx = fg -\int f'g dx

Identify suitable f and g in the original expression. I use:

f=x\,\,\,, \,\,\,g=\sqrt{1-x}\\f'=1\,\,\,,\,\,\,g'=-\frac{2}{3}(1-x)^\frac{3}{2}

and write the integration by parts :

\int x\sqrt{1-x} dx = -\frac{2}{3}x(1-x)^\frac{3}{2}+\frac{2}{3}\int (1-x)^\frac{3}{2}dx=\\ =-\frac{2}{3}x(1-x)^\frac{3}{2}-\frac{4}{15}(1-x)^\frac{5}{2}+C=\\=-\frac{2}{15}(1-x)^\frac{3}{2}(3x+2)+C

with C an arbitrary integration constant.

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