Question

Find or evaluate the integral. (Complete the square, if necessary.) $$ \int \frac{x}{\sqrt{9+8 x^{2}-x^{4}}} d x $$

Short answer

The integral evaluates to \( \frac{2}{5} \ln|\sin(\arctan(\frac{2}{5}(x^2 - \frac{4}{3})))| + C \).

Step-by-step solution

Completing the square for the denominator of the integrand

Identify the highest degree of $x$ in the denominator, which is $x^4$. To complete the square in the denominator, rewrite the quadratic in the form $(x^2-a)^2-b^2$ where $a$ and $b$ are constants. Hence, the integrand becomes: \( \frac{x}{\sqrt{(x^{2}-\frac{4}{3})^2-\frac{25}{9}}} \)

Applying trigonometric substitution

To convert the integrand into a form that can be more easily integrated, let $x^2 - \frac{4}{3} = \frac{5}{3}\tan^2(\theta)$. Then, $\sqrt{(x^{2}-\frac{4}{3})^2-\frac{25}{9}} = \frac{5}{3}|\tan(\theta)|$. Also, $2x dx = \frac{10}{3}\sec^2(\theta) d\theta$. So, replace $x dx$ with $\frac{5}{6}\sec^2(\theta) d\theta$, and substitute these values into the integral equation.

Simplifying and Evaluating the Integral

The integral becomes \( \int \frac{\frac{5}{6}\sec^2(\theta)}{\frac{5}{3}|\tan(\theta)|} d\theta \). Simplifying this gives \( \int \frac{2}{5} |\cot(\theta)| d\theta = \frac{2}{5} \ln|\sin(\theta)| + C \). Then resubstitute $ \theta = \arctan(\frac{2}{5}(x^2 - \frac{4}{3}))$ to get the expression back in terms of $x$: \( \frac{2}{5} \ln|\sin(\arctan(\frac{2}{5}(x^2 - \frac{4}{3})))| + C \)