1. integration arcsin rule

$\int \frac{7x}{\sqrt{16-x^4}} dx = 7\int \frac{x}{\sqrt{4^2-(x^2)^2}} dx$

The arcsin rule can be used here, right? I'm not sure what to do with the x in the numerator to be able to apply the arcsin rule.

2. Originally Posted by dataspot
$\int \frac{7x}{\sqrt{16-x^4}} dx = 7\int \frac{x}{\sqrt{4^2-(x^2)^2}} dx$

The arcsin rule can be used here, right? I'm not sure what to do with the x in the numerator to be able to apply the arcsin rule.
First, you will need to use the substitution of $u = x^2$ to obtain $\frac 72 \int \frac 1{\sqrt{16 - u^2}} \ \mathrm{d}u$.

Then you can apply the arcsin rule which is $\int \frac{1}{\sqrt{a^2 - x^2}} \ \mathrm{d}x = \mathrm{arcsin}\left(\frac{x}{a}\right)$.

Remember to change the $u$ back to $x^2$ in the final answer.

3. Originally Posted by Air
First, you will need to use the substitution of $u = x^2$ to obtain $\frac 72 \int \frac 1{\sqrt{16 - u^2}} \ \mathrm{d}u$.

Then you can apply the arcsin rule which is $\int \frac{1}{\sqrt{a^2 - x^2}} \ \mathrm{d}x = \mathrm{arcsin}\left(\frac{x}{a}\right)$.

Remember to change the $u$ back to $x^2$ in the final answer.
Got it, thanks!