Evaluate the indefinite integral. Use the Substitution Rule.?

Question

The integral of x^2 / sqrt(1-x) dx


Please show your work. Thank you. ;)

Answer 1

x² / √(1-x)
= x² √(1-x)/(1 - x)
= (-1 + 1 - x²)√(1-x)/(1 - x)
= (1 - x²)√(1-x)/(1 - x) - √(1-x)/(1 - x)
= (1 + x)√(1-x) - √(1-x)/(1 - x)
= x√(1-x) + √(1-x) - 1/√(1-x)

So ∫x² / √(1-x) . dx
= ∫[x√(1-x) + √(1-x) - 1/√(1-x)] . dx

Now ∫x√(1-x)dx
= x * -1 * 2/3 (√(1-x)³ - ∫[-1 * 2/3 (√(1-x)³ . 1)dx
= -2x/3 * √(1-x)³ + 2/3∫(√(1-x)³ dx
= -2x/3 * √(1-x)³ + 2/3*2/5 * -1 * √((1-x)^5)
= -2x/3 * √(1-x)³ - 4/15 * √((1-x)^5)

∫√(1-x) dx =

∫1/√(1-x) dx = -2 *√(1-x)

So ∫x² / √(1-x) . dx
= ∫[x√(1-x) + √(1-x) - 1/√(1-x)] . dx
= -2x/3 * √(1-x)³ - 4/15 * √((1-x)^5) - 2/3 * √(1-x)³ + 2 *√(1-x) + c
= - 4/15 * √((1-x)^5) -2/3 (x + 1)√(1-x)³ + 2 *√(1-x) + c

Answer 2

When using the substitution rule, try substituting to get rid of the most ugly part of the expression. In this case its sqrt(1-x). So let y = sqrt(1-x). Then y^2 = 1-x, so x = 1-y^2.
Now, its a bit easier calculating dx/dy than dy/dx, so dx/dy = -2y. So dx = -2ydy.
Finally, we need the x^2 on the top, which is (1-y^2)^2.
Putting that all together, we want the integral of:
((1-y^2)^2)/y * -2y dy
The ys cancel out to give integral of -2(1-y^2)^2.
We can expand that to get -2y^4 + 4y^2 - 2.
Now thats easy to integrate; its -2/5 y^5 + 4/3 y^3 - 2y + c.
Finally, substitute the x back in: -2/5 (sqrt(1-x))^5 + 4/3 (sqrt(1-x))^3 - 2(sqrt(1-x)) + c.
I could easily have made a typo in all that, but you get the general idea :)

Answer 3

This integral is an example of one involving a linear
irrationality.
So let u= 1-x, x = 1-u, dx = -du.
So we get
-INT [u² -2u +1)/u^1/2 du].
But this is easy to integrate by the power rule.
The answer is therefore
-(2/5*u^5/2 -4/3*u^(3/2) + 1/2*u^(1/2) )+ C
Finally, replace u by 1-x to get the final answer.
Hope that helps a bit.

Answer 4

you start up with the aid of recognizing that y^3 is the by-product (y^4)/4; So in case you place u = (2*y^4 - one million), then du = 8*y^3; take the ingredient of 8 exterior the fundamental to get (one million/8)?[one million/?u]du; The fundamental is one million/8 * 2*?u = one million/4 * ?[2*y^4 - one million] + C you could examine that it somewhat is right with the aid of taking the by-made of one million/4 * ?[2*y^4 - one million] + C