Question

Evaluate the following integrals : $$\int \frac{\left(x^{2}-1\right) d x}{x \sqrt{x^{4}+3 x^{2}+1}}$$

Short answer

Short Answer: The integral of $$\int \frac{x^2 - 1}{x\sqrt{x^4+3x^2+1}} dx$$ is equal to $$\frac{1}{2}\left((-x^2 + 2) \frac{-1}{\sqrt{x^4+3x^2+1}}\right) - \frac{\sqrt{x^4+3x^2+1}}{3} x^{2} + C$$, where C is the integration constant.

Step-by-step solution

Apply Substitution

Let's apply a substitution such that: $$u = x^2 + 1$$ Then differentiate both sides: $$\frac{du}{dx} = 2x$$ So, $$dx = \frac{du}{2x}$$ Now, substitute these expressions into the original integral: $$\int \frac{(x^2-1) \frac{du}{2x}}{x \sqrt{u^2 + 2u}}$$

Simplify the Integral

We can simplify the integral by cancelling out the common terms: $$\frac{1}{2} \int \frac{(u-2) du}{u \sqrt{u^2 + 2u}}$$

Apply Integration by Parts

Now we'll apply integration by parts to the simplified integral. Let: $$v = u-2$$ $$dv = du$$ Also, let $$w = \frac{1}{u \sqrt{u^2 + 2u}}$$ Now differentiate v and integrate w: $$dw = \frac{-1}{u^2 \sqrt{u^2 + 2u}} du$$ Now use the formula for integration by parts: $$\int vw = vw - \int w dv$$ Substituting the variables: $$\frac{1}{2}\left(\int (u-2) \frac{1}{u\sqrt{u^2+2u}} du = (u-2) \frac{-1}{\sqrt{u^2+2u}} - \int \frac{-1}{\sqrt{u^2+2u}} du\right)$$

Evaluate the Integral

Now we can evaluate the remaining integral: $$\int \frac{-1}{\sqrt{u^2+2u}} du = - \int \frac{1}{\sqrt{1+\frac{2}{u}}} \frac{du}{\sqrt{u}}$$ Now once again use the substitution: $$t = 1 + \frac{2}{u}$$ So, $$dt = \frac{-2}{u^2} du$$ Substitute these expressions into the integral: $$- \int \frac{1}{\sqrt{t}} \frac{-u^2}{2} dt$$ Now we can simplify the integral and obtain: $$\frac{1}{2} \int \frac{u^2}{\sqrt{t}} dt$$ Now, we can integrate: $$\frac{1}{2} \int \frac{u^2}{\sqrt{t}} dt = \frac{1}{2} \left(-\frac{2 \sqrt{t}}{3} u^{2}\right)$$ Now substitute back the variables: $$- \frac{\sqrt{1+\frac{2}{u}}}{3} u^{2}$$ Finally, substitute back the original variable x: $$- \frac{\sqrt{x^4+3x^2+1}}{3} x^{2}$$

Write the Final Answer

Combine the results from steps 3 and 4 to form the final answer: $$\frac{1}{2}\left((-x^2 + 2) \frac{-1}{\sqrt{x^4+3x^2+1}}\right) - \frac{\sqrt{x^4+3x^2+1}}{3} x^{2} + C$$ Where C is the integration constant.