Question

I am given an integral to solve with given substitution values. I got an answer, but I'm not quite sure if it's correct as an online integral calculator gave a different answer.

∫ x sqrt(4-x) dx

Given that u = 4-x .

In this case, x = 4 - u

du = -dx

Now..

=- ∫ (4-u)sqrt(u) du
=- ∫ (4-u)u^(1/2) du

=- ∫ 4u^(1/2) - u^(3/2) du

=-4∫ u^(1/2) - u^(3/2) du

=-4((2/3)u^(3/2) - (2/5)u^(5/2))

=-4((2/3)(4-x)^(3/2) - (2/5)(4-x)^(5/2)) + C is my answer.

Is there anything wrong with my math during the process? Any help is greatly appreciated!

Answer 1

=- ∫ (4-u)sqrt(u) du
=- ∫ (4-u)u^(1/2) du
***= -4∫u^.5du + ∫u^1.5 du
***= -4(u^1.5/1.5) + u^2.5/2.5 + c
***= -(8/3)u^(3/2) + (2/5)u^(5/2)+c

we disagree about sign of second term

=- ∫ 4u^(1/2) - u^(3/2) du

=-4∫ u^(1/2) - u^(3/2) du

=-4((2/3)u^(3/2) - (2/5)u^(5/2))

=-4((2/3)(4-x)^(3/2) - (2/5)(4-x)^(5/2)) + C is my answer.