integration homework help

**asilvester635** · December 11th 2012, 10:15 AM

I rewrote this as

= t(t - 4)^1/3.... do I combine them and integrate after?

**Zouzz** · December 11th 2012, 10:35 AM

Integration by parts, like the other one i commented on u(x)=(t-4)^(1/3), v(x)=t

**skeeter** · December 11th 2012, 10:37 AM

$\int t \sqrt[3]{t-4} \, dt$

$u = t-4$

$t = u+4$

$du = dt$

substitute ...

$\int (u+4) \cdot \sqrt[3]{u} \, du$

can you finish?

**Zouzz** · December 11th 2012, 10:41 AM

...or that XD

**asilvester635** · December 11th 2012, 10:56 AM

this is different on how i learned it.... so i integrate the right side??? sorry idk x/

**asilvester635** · December 11th 2012, 12:01 PM

wait i got it... im working on it at the moment....
okay heres what i've got so far

u = t - 4
du = dt
t = u + 4

= ^3√u ( u + 4) + c
= u^1/3 (u + 4) + c

now do i multiply u^1/3 (u + 4) together and integrate?

**Prove It** · December 11th 2012, 02:44 PM

Yes.

**x3bnm** · December 19th 2012, 05:20 AM

$\text{We know that } \int u \, dv=uv-\int v \, du.\!$

So for $\int t\sqrt[3]{t-4} \;\;dt$ let $u = t \therefore du = dt$
And let $dv = \sqrt[3]{t-4} \;\;dt \therefore v = \frac{3}{4}(t - 4)^{\frac{4}{3}}$

$\begin{align*}\int t\sqrt[3]{t-4} \;\;dt =& \frac{3}{4}t(t-4)^{\frac{4}{3}} - \int \frac{3}{4}(t - 4)^{\frac{4}{3}} \\ =& \frac{3}{4}t(t-4)^{\frac{4}{3}} - \frac{9}{28}(t-4)^{\frac{7}{3}} + C \\ =& \frac{3}{7}(t-4)^{\frac{4}{3}}[\frac{7t}{4} - \frac{3t}{4} + \frac{12}{4}] + C \\ =& \frac{3}{7}(t-4)^{\frac{4}{3}}(t+3) + C\end{align*}$

$\therefore \int t\sqrt[3]{t-4} \;\;dt = \frac{3}{7}(t-4)^{\frac{4}{3}}(t+3) + C$

Check:

int x(x-4)^(1/3) dx - Wolfram|Alpha

**hollywood** · December 19th 2012, 11:10 AM

Or starting from skeeter's post,

$\int (u+4) \cdot \sqrt[3]{u} \, du =\int u^\frac{4}{3} + 4u^\frac{1}{3} \, du =$

$\frac{3}{7}u^\frac{7}{3}+ 3 u^\frac{4}{3} + C = \frac{3}{7}u^\frac{4}{3} (u + 7) + C=$

$\frac{3}{7} (t-4)^\frac{4}{3} (t+3) + C$

- Hollywood

**vicwilson123** · March 29th 2013, 04:53 AM

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**HallsofIvy** · March 29th 2013, 06:56 AM

Originally Posted by asilvester635

wait i got it... im working on it at the moment....
okay heres what i've got so far

u = t - 4
du = dt
t = u + 4

= ^3√u ( u + 4) + c
= u^1/3 (u + 4) + c

now do i multiply u^1/3 (u + 4) together and integrate?

You have not yet integrated so there should be no "+ c".

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