Please help with antiderivatives?

Question

evaluate the integral by rewriting the integral appropiately and if necessary apply the power rule:

x^3 √x dx = the answer is 2/9x^9/2 + c but i need to know why i don't know how to solve it please can somebody explain?

Answer 1

∫x³*√x dx, is this correct?

multiply and exponents add

∫x^3/2 dx

integrate

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

2/5x^5/2

which doesn't match your answer, so I suspect you are mis stating the question.

I REALLY HATE GUESSING AT THE QUESTION just because you didn't check it!

try ∫x(³√x) dx

∫x^(1+1/3) dx = ∫x^(4/3) dx

integrate

(3/7) x^(7/3)

doesn't match either.

give up

Answer 2

ok, here's how you do it

you know that integral(pq') = pq-integral(p'q)

so you have to get p and q in order to solve it

let's say

p= x^(1/2)

p'=x^(-1/2)/2

q=x^4/4

q'=x^3

as you may see p and q' are parts of the original equation. So

integral(x^3 √x dx)=x^4*x^(1/2)/4 - integral(x^-(1/2)/2*x^4/4)

=x^(9/2)/4-integral(x^(7/2)/8)

=x^(9/2)/4-2*x^(9/2)/78

=x^(9/2)/4-x^(9/2)/36

=2/9*x^(9/2) + C

i'm sorry i didn't explain more because i'm in hurry, but hope this helps