Integral question $\int x \sqrt {1-x}\ dx.$

So there are (from my knowledge) $$2$$ ways of solving for $$\int x \sqrt {1-x}\ dx.$$

The first is by $$u$$ substitution and the second is by parts. They both differ now i'm confused which one is correct? I have both the correct answers but they differ?

• Differentiate both results – lab bhattacharjee Apr 7 at 11:27
• Both are correct, of course. You should edit your question, showing us your work so far. – José Carlos Santos Apr 7 at 11:29
• I got this here $$-\frac{2}{15} (1-x)^{3/2} (3 x+2)+C$$ – Dr. Sonnhard Graubner Apr 7 at 11:31
• Subtract one result from the other. Simplify as much as possible. Your answer should be a constant. – Bernard Massé Apr 7 at 11:36

Let me guess: you substituted $$u=1-x$$ to get $$\int(u^{3/2}-u^{1/2})du=\frac{2}{5}(1-x)^{5/2}-\frac{2}{3}(1-x)^{3/2}+C_1$$, but for parts you wrote $$f=x,\,g=-\frac{2}{3}(1-x)^{3/2}$$ to get $$-\frac{2}{3}x(1-x)^{3/2}+\frac{2}{3}\int(1-x)^{3/2}dx=-\frac{2}{3}x(1-x)^{3/2}-\frac{4}{15}(1-x)^{5/2}+C_2.$$But from $$x(1-x)^{3/2}=(1-x)^{3/2}-(1-x)^{5/2}$$, these results are identical (because $$\frac{2}{3}-\frac{4}{15}=\frac{2}{5}$$) with $$C_1=C_2$$. (Sometimes when you calculate an indefinite integral by multiple methods, the integration constants differ, but that's fine.)

• Thank you, this is exactly how i worked it out. I did not realize that i had to get rid of the x in 2/3x(1−x)^3/2 by using u substitution. That was pretty tricky. – Shaun Weinberg Apr 7 at 12:16

You should find that you can reduce it to a single constant (which may be zero or non-zero). This is expected. The indefinite integral is always expressed with a "$$+c$$" at the end of the answer - an arbitrary constant of integration. There is no one answer to an indefinite integral, the answer is an infinite family of expressions, all separated from each other by every conceivable constant.
$$x\sqrt{1-x}=(x-1+1)\sqrt{1-x}=-(1-x)\sqrt{1-x}-\sqrt{1-x}$$ $$=-(1-x)^{\frac{3}{2}}-(1-x)^{\frac{1}{2}}$$ The anti derivative of $$(1-x)^a$$ is $$-\frac{1}{a+1}(1-x)^{a+1}, a\ne -1$$