1. ## Integral x^4(4+x^5)^1/2

$\displaystyle \int x^4\sqrt{4+x^5}$

Do I make $\displaystyle u=4+x^5$ in order to solve this problem?

2. Originally Posted by cammywhite
$\displaystyle \int x^4\sqrt{4+x^5}$

Do I make $\displaystyle u=4+x^5$ in order to solve this problem?
Yes! Just remember to factor $\displaystyle \frac{1}{5}$ out of your integral!

3. Originally Posted by cammywhite
$\displaystyle \int x^4\sqrt{4+x^5}$

Do I make $\displaystyle u=4+x^5$ in order to solve this problem?
Yes.

Notice that if $\displaystyle u = 4 + x^5$ then $\displaystyle \frac{du}{dx} = 5x^4$.

So $\displaystyle \int{x^4\sqrt{4 + x^5}\,dx} = \frac{1}{5}\int{5x^4(4 + x^5)^{\frac{1}{2}}\,dx}$

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,\frac{du}{dx}\,dx }$

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,du}$.

I trust you can go from here...

4. Originally Posted by Prove It
Yes.

Notice that if $\displaystyle u = 4 + x^5$ then $\displaystyle \frac{du}{dx} = 5x^4$.

So $\displaystyle \int{x^4\sqrt{4 + x^5}\,dx} = \frac{1}{5}\int{5x^4(4 + x^5)^{\frac{1}{2}}\,dx}$

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,\frac{du}{dx}\,dx }$

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,du}$.

I trust you can go from here...

Let me try

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,du}$
$\displaystyle = \frac{1}{5}\int\frac{{u^{\frac{3}{2}}\,du}}{\frac{ 3}{2}}$

so far correct?

5. Originally Posted by cammywhite
Let me try

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,du}$
$\displaystyle = \frac{1}{5}\int\frac{{u^{\frac{3}{2}}\,du}}{\frac{ 3}{2}}$

so far correct?
Your integration is correct, but you need to leave out the integral sign & du since you have already integrated.

6. Originally Posted by mollymcf2009
Your integration is correct, but you need to leave out the integral sign & du since you have already integrated.
I looked at the answer and it's $\displaystyle \frac {\sqrt{x^5+4}(2x^5+8)}{15}$

I have no clue how to get to the answer

7. Originally Posted by cammywhite
Let me try

$\displaystyle = \frac{1}{5}\int{u^{\frac{1}{2}}\,du}$
$\displaystyle = \frac{1}{5}\int\frac{{u^{\frac{3}{2}}\,du}}{\frac{ 3}{2}}$

so far correct?
$\displaystyle \frac{1}{5} \left( \frac{u^{3/2}}{\frac{3}{2}}\right)=\frac{2}{15}u^{3/2}$

$\displaystyle \frac{2}{15}(4+x^5)^{3/2}=\frac{2}{15}\sqrt{4+x^5}(4+x^5)=\frac{2(4+x^5)\ sqrt{4+x^5}}{15}=\frac{(8+2x^5)\sqrt{4+x^5}}{15}$

8. Originally Posted by TheEmptySet
$\displaystyle \frac{1}{5} \left( \frac{u^{3/2}}{\frac{3}{2}}\right)=\frac{2}{15}u^{3/2}$

$\displaystyle \frac{2}{15}(4+x^5)^{3/2}=\frac{2}{15}\sqrt{4+x^5}(4+x^5)=\frac{2(4+x^5)\ sqrt{4+x^5}}{15}=\frac{(8+2x^5)\sqrt{4+x^5}}{15}$

i don't get this part

$\displaystyle \frac{2}{15}(4+x^5)^{3/2} =\frac{2}{15}\sqrt{4+x^5}(4+x^5)$

9. Originally Posted by cammywhite
i don't get this part

$\displaystyle \frac{2}{15}(4+x^5)^{3/2} =\frac{2}{15}\sqrt{4+x^5}(4+x^5)$
We are using this exponent law

$\displaystyle x^ax^b=x^{a+b}$

Note that $\displaystyle x^{\frac{3}{2}}=x^{\frac{1}{2}}x^1=\sqrt{x}\cdot x$

So using this we get

$\displaystyle (4+x^5)^{3/2} =\sqrt{4+x^5}(4+x^5)$

10. Integration by substitution is like a backward-application of the chain rule. In the expression

$\displaystyle x^4\sqrt{4+x^5},$

we see that $\displaystyle x^4$ looks like it was differentiated from $\displaystyle 4+x^5$ of the square root.

To find the real antiderivative, we just add in the factor $\displaystyle \frac{1}{5}$ to get $\displaystyle x^4$ from $\displaystyle 5x^4$.