Integration by parts: int [x sqrt x - 1] dx

[mooshupork34]

ok so i had to compute the integral of x * the square root of x-1 by parts.

this is what i did:

i set u = x, v' = (x-1)^(1/2), u' = 1, and v = 2/3(x-1)^(3/2)

then i got:

2/3 x (x - 1)^(3/2) - 2/3 * the integral of (x-1)^3/2 dx

2/3 x (x-1)^(3/2) - 2/3 * the integral of 2/5 (x-1)^(5/2) + B

my question is, how do you simplify the above answer? i'm having trouble with it.

 

[skeeter]

why parts, when change of variables is so much easier?

\(\displaystyle \L \int x\sqrt{x-1} dx\)

\(\displaystyle \L u = x-1\)
\(\displaystyle \L x = u+1\)
\(\displaystyle \L du = dx\)

\(\displaystyle \L \int (u+1)\sqrt{u} du\)

\(\displaystyle \L \frac{2}{5}u^{\frac{5}{2}} + \frac{2}{3}u^{\frac{3}{2}} + C\)

\(\displaystyle \L \frac{2}{15}u^{\frac{3}{2}} [3u + 5] + C\)

\(\displaystyle \L \frac{2}{15}(x-1)^{\frac{3}{2}} [3(x-1) + 5] + C\)

\(\displaystyle \L \frac{2}{15}(x-1)^{\frac{3}{2}}(3x+2) + C\)

 

[galactus]

Skeeter, as always, is correct. But, if you must use parts:

\(\displaystyle \L\\u=x, \;\ dv=\sqrt{x-1}dx, \;\ du=dx, \;\ v=\frac{2}{3}(x-1)^{\frac{3}{2}}\)

\(\displaystyle \L\\\frac{2x}{3}(x-1)^{\frac{3}{2}}-\int\frac{2}{3}(x-1)^{\frac{3}{2}}dx\)

\(\displaystyle \L\\\frac{2x}{3}(x-1)^{\frac{3}{2}}-\frac{4}{15}(x-1)^{\frac{5}{2}}\)

Factor to tidy up:

\(\displaystyle \L\\\frac{2}{3}(x-1)^{\frac{3}{2}}(x-\frac{2}{5}(x-1))\)

or

\(\displaystyle \L\\\frac{2(x-1)^{\frac{3}{2}}(3x+2)}{15}\)

 

[mooshupork34]

Thanks!

Just one question, though.

How does one get from 2x/3 (x-1)^(3/2) - 4/15 (x-1)^(5/2)

to

2(x-1)^(3/2) * (3x+2)
divided by 15

 

[skeeter]

Thanks!

Just one question, though.

How does one get from 2x/3 (x-1)^(3/2) - 4/15 (x-1)^(5/2)

to

2(x-1)^(3/2) * (3x+2)
divided by 15

\(\displaystyle \L \frac{2x}{3} (x-1)^{\frac{3}{2}} - \frac{4}{15} (x-1)^{\frac{5}{2}}\)

the two terms have \(\displaystyle \L \frac{2}{3}(x-1)^{\frac{3}{2}}\) in common, factor it out ...

\(\displaystyle \L \frac{2}{3}(x-1)^{\frac{3}{2}} \left[x - \frac{2}{5}(x-1)\right]\)

\(\displaystyle \L \frac{2}{3}(x-1)^{\frac{3}{2}} \left(\frac{3x}{5} + \frac{2}{5}\right)\)

\(\displaystyle \L \frac{2}{15}(x-1)^{\frac{3}{2}}(3x + 2)\)