# Any hints on how to compute this integral?

Could anyone please give me a hint on how to compute the following integral?

$$\int \sqrt{\frac{x-2}{x^7}} \, \mathrm d x$$

I'm not required to use hyperbolic/ inverse trigonometric functions.

• There exists and elementary antiderivative, if helpful. – Brian Apr 1 '19 at 17:24
• Are you required to not use them? – eyeballfrog Apr 1 '19 at 17:56
• Yeah, unfortunately. @Brian yeah but i need to see the steps. – Just_Cause Apr 1 '19 at 18:04

Write your integrand in the form $$\frac{\sqrt{x-2}}{x^{7/2}}$$ and then substitute $$u=\sqrt{x}$$ so you will get $$2\int\frac{\sqrt{u^2-2}}{u^6}du$$ after this substitute $$u=\sqrt{2}\sec(s)$$ to get $$2\sqrt{2}\int\frac{\sin^2(s)\cos^2(s)}{4\sqrt{2}}ds$$

• You made sqrt(x) a function of s? Could you elaborate more please? – Just_Cause Apr 1 '19 at 18:06

Write $$y(x):=\sqrt{\frac {x-2} {x^7}}$$.

Note that $$y'(x)= \frac {7-3x}{x^8} \frac 1 y$$

Hence $$\frac d {dx} x^n y=n x^{n-1} y + x^{n-8} \frac {7-3x} y.$$

Do the ansatz $$F(x)=\sum_{n=0}^k a_nx^ny ~~~~\text{ and } ~~~~F'(x)=y(x).$$

We get $$y\sum_{n=1}^{k}a_n nx^{n-1} +\frac 1 y\sum_{n=0}^{k} a_n x^{n-8} ({7-3x})=y$$

Thus $$\sum_{n=0}^{k} a_n x^{n-8} ({7-3x})=y^2(1-\sum_{n=1}^{k}a_n nx^{n-1}).$$

Now insert $$y^2$$ and get $$({7-3x})\sum_{n=0}^{k} a_n x^{n-1} = (x-2 )(1-\sum_{n=1}^{k}a_n nx^{n-1})$$ or equivalently $$2-x+({7-3x})\sum_{n=0}^{k} a_n x^{n-1}+ (x-2)\sum_{n=1}^{k}a_n nx^{n-1}=0.$$

We solve this recursively.

The lowest order is $$x^{-1}$$. There we have $$7a_0 x^{-1}=0$$, so $$a_0=0$$.

In constant order: $$2+7a_1-3a_0-2a_1=0$$, so $$a_1=-\frac 2 5$$

In order $$x$$: $$(-1+7a_2 -3 a_1 +a_1-4 a_2)x=0$$ , so $$a_2= \frac 1 3 (1+2 a_1)=\frac 1 {15}$$

in order $$x^2$$: $$(7a_3-3a_2+2a_2-6 a_3)x^2=0$$, so $$a_3=a_2=\frac 1 {15}$$.

in order $$x^3$$: $$(7a_4-3a_3+3a_3 - 8a_4)x^4=0$$, so $$a_4=0$$

in order $$x^n$$ for $$n>3$$: $$(7a_{n+1} - 3 a_n +n a_n - 2(n+1) a_{n+1})x^n$$, so $$a_{n+1}=\frac {n-3}{7-2(n+1)} a_n=0$$.

In conclusion it follows that $$\int y(x)= const + F(x)= const+ \frac 1 {15} (-6x+ x^2+x^3) y$$

With $$y:=\dfrac1x$$,

$$\int\sqrt{\frac{x-2}{x^7}}dx=-\int\sqrt{\left(\frac1y-2\right)y^7}\,\frac{dy}{y^2}=-\int y\sqrt{1-2y}\,dy.$$

Then by parts,

$$-\int y\sqrt{1-2y}\,dy=\frac13y(1-2y)^{3/2}-\frac13\int(1-2y)^{3/2}dy=\frac13y(1-2y)^{3/2}+\frac1{15}(1-2y)^{5/2}$$

$$=\frac1{3x}\left(1-\frac2x\right)^{3/2}+\frac1{15}\left(1-\frac2x\right)^{5/2}.$$

• How could you get the y^6 out of the sqrt without facing issues with absolute value? – Just_Cause Apr 2 '19 at 16:58
• @Just_Cause: I didn't worry about the domain of validity of the development. This must be added, with a thorough discussion, as a "post processing". – Yves Daoust Apr 2 '19 at 17:53
• Yeah, I'll have this topic discussed ASAP. – Just_Cause Apr 3 '19 at 16:33
• @YvesDaoust Oh, I think what he means is that you should have $\sqrt {x^6}=\sqrt {(x^3)^2}=|x^3|=|x||x^2|=x^2|x|=x^3$ only for $x\ge 0.$ Thus, your substitution in actual fact yields $\int{|y|\sqrt{1-2y}}\mathrm{d}y.$ This is what you should have computed. You only did the case when $|y|=y.$ The other case yields an improper integral since then $y$ becomes unbounded as $x\to0^-.$ Thus, it has to be dealt with specially. – Allawonder May 1 '19 at 18:27

Try the substitution $$u=\frac{x-2}{x}$$

or equivalently

$$x=\frac{2}{1-u}$$

• Tried before, I'd face problems with absolute values cuz I get an even power under the sqrt – Just_Cause Apr 1 '19 at 18:07