Question on Integration?

Question

I am in my second try of Calculus 2 and I am having issues with this integration problem maybe someone here can just give me the final answer so I can work to it, or even the steps to get to it. I just really need to get this question done as I have been racking my brain on it for a little under an hour.

It's the integral of X^4(Square Root of: 10+X^5)dx

I belive it is a subsitution problem. Any help would be great.

Answer 1

You don't have to do it with substution, but sometimes it's just easier to do that in order to see it more clearly. Let u = 10+x^5, then du = 5x^4 dx. So 1/5 du = x^4 dx.

In terms of u,

1/5 ⌡u^.5 du

(1/5)(2/3) u^1.5 + C
= (2/15) (10+x^5)^1.5 + C

Answer 2

Okay, the first thing you have to do is rewrite the problem:

The integral of (10 + x^5)^(1/2) x^4 dx

u = 10 + x^5
du = 5x^4 dx

You have to multiply the whole integral by the reciprocal of 5 because it doesn't completely match the dx.

Therefore:

(1/5) Integral of: (10 + x^5)^(1/2) 5x^4 dx

Then you can take out the Integral sign and the 5 and find the anti-derivative for the (10 + x^5)^(1/2).

(1/5) (2/3) (10 + x^5)^(3/2) + C

All you have to do to solve now is multiply the (1/5)(2/3) (I would but I don't have my calc. and I wouldn't want to tell you wrong.) to get your answer. Hope this helps!

(1/5)

Answer 3

It isn't substitution, it can just be done by sight

(x^4)*(10+x^5)^(1/2)

by reversing the chain rule you know will need to start with the something in the form a(10+x^5)^(3/2)

if this is differentiated then you get a(15/2)*(x^4)*(10+x^5)^(1/2)

from the original problem we know that a(15/2) = 1
therefore a = 2/15

so the integral is (2/15)*(10+x^5)^(3/2) + C

Answer 4

integral x^4 sqrt( 10+x^5) dx
= 2/15 integral (3/2)sqrt(10+x^5)(5x^4)dx
= 2/15 integral (3/2)sqrt(10+x^5) d(x^5)
= 2/15 (10 + x^5)^(3/2) + c

(in effect you are make a u = x^5 substitution, but you would normally just do in your head)

Answer 5

Assume that question is :-
I = ∫ x^(4).(10 + x^(5))^(1/2).dx
Let u = 10 + x^(5)
du = 5.x^(4).dx
du/5 = x^(4).dx
I = (1/5).∫ u^(1/2).du
I = (1/5).u^(3/2) / (3/2) + C
I = (2/15).(10 + x^(5))^(3/2) + C