Thread: Integration with u-sub and trig

1. Integration with u-sub and trig

$\int \frac{x^4}{\sqrt{x^{10}-2}}dx$

Let: $u=x^5 \text{ }du=5x^4dx$

$=\tfrac{1}{5}\int\frac{1}{\sqrt{u^2-(\sqrt{2})^2}}du$

Let: $u=\sqrt{2}\sec\theta \text{ }du=\sqrt{2}\sec\theta\tan\theta d\theta$

$=\tfrac{1}{5}\int\frac{\sqrt{2}\sec\theta\tan\thet a}{\sqrt{2}\tan\theta}d\theta$
$=\tfrac{1}{5}\int\sec\theta d\theta$
$=\tfrac{1}{5}\ln{\left|\sec\theta+\tan\theta\right |}+C$
$=\tfrac{1}{5}\ln{\left|\tfrac{u}{\sqrt{2}}+\tfrac{ \sqrt{u^2-2}}{\sqrt{2}}\right|}+C$
$=\tfrac{1}{5}\ln{\left|\tfrac{x^5}{\sqrt{2}}+\tfra c{\sqrt{x^{10}-2}}{\sqrt{2}}\right|}+C$

The book claims that the answer is:
$=\tfrac{1}{5}\ln{\left|x^5+\sqrt{x^{10}-2}\right|}+C$

I could see where the discrepancy could come from, but I don't see the error in my work... what did I do wrong?

2. Hello !
Originally Posted by symstar
$\int \frac{x^4}{\sqrt{x^{10}-2}}dx$

Let: $u=x^5 \text{ }du=5x^4dx$

$=\tfrac{1}{5}\int\frac{1}{\sqrt{u^2-(\sqrt{2})^2}}du$

Let: $u=\sqrt{2}\sec\theta \text{ }du=\sqrt{2}\sec\theta\tan\theta d\theta$

$=\tfrac{1}{5}\int\frac{\sqrt{2}\sec\theta\tan\thet a}{\sqrt{2}\tan\theta}d\theta$
$=\tfrac{1}{5}\int\sec\theta d\theta$
$=\tfrac{1}{5}\ln{\left|\sec\theta+\tan\theta\right |}+C$
$=\tfrac{1}{5}\ln{\left|\tfrac{u}{\sqrt{2}}+\tfrac{ \sqrt{u^2-2}}{\sqrt{2}}\right|}+C$
$=\tfrac{1}{5}\ln{\left|\tfrac{x^5}{\sqrt{2}}+\tfra c{\sqrt{x^{10}-2}}{\sqrt{2}}\right|}+C$

The book claims that the answer is:
$=\tfrac{1}{5}\ln{\left|x^5+\sqrt{x^{10}-2}\right|}+C$

I could see where the discrepancy could come from, but I don't see the error in my work... what did I do wrong?
It is correct

$\tfrac{x^5}{\sqrt{2}}+\tfrac{\sqrt{x^{10}-2}}{\sqrt{2}}=\tfrac{1}{\sqrt{2}} \left(x^5+\sqrt{x^{10}-2}\right)$

Thus $\tfrac{1}{5}\ln{\left|\tfrac{x^5}{\sqrt{2}}+\tfrac {\sqrt{x^{10}-2}}{\sqrt{2}}\right|}+C=\tfrac 15 \ln\left|\tfrac{1}{\sqrt{2}} \left(x^5+\sqrt{x^{10}-2}\right)\right|+C$

Use the rule $\ln(ab)=\ln(a)+\ln(b)$ :

$=\tfrac 15 \left(-\ln(\sqrt{2})+\ln\left|x^5+\sqrt{x^{10}-2}\right)\right|+C$

$=\tfrac 15 \ln\left|x^5+\sqrt{x^{10}-2}\right|\underbrace{-\tfrac{\ln(\sqrt{2})}{5}+C}_{\text{this is a constant}}$

$=\tfrac 15 \ln\left|x^5+\sqrt{x^{10}-2}\right|+C'$

3. Ah, I see. Thanks!