Question

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

Short answer

#Revised Solution# #tag_title# Step 1: Make a more suitable substitution #tag_content# After analyzing the problem, we realize we need a more suitable substitution to simplify the integral. Let's try the substitution: $$x = \sinh(y)$$ Now, differentiate 'x' with respect to 'y': $$\frac{dx}{dy} = \cosh(y)$$ Solve for \(dy\): $$dy = \frac{dx}{\cosh(y)}$$ #tag_title# Step 2: Replace x and dx in the integral #tag_content# Now substitute the expressions for \(x\) and \(dy\) into the integral: $$\int \frac{(1+\sinh^2(y))}{(1-\sinh^2(y))} \frac{dx}{\sqrt{1+\sinh^2(y)+\sinh^4(y)}\cosh(y)}$$ #tag_title# Step 3: Apply hyperbolic identity to simplify #tag_content# Apply the hyperbolic identity \(1 + \sinh^2(y) = \cosh^2(y)\) to the integral: $$\int \frac{\cosh^2(y)}{\cosh^2(y)-\sinh^2(y)} \frac{dx}{\sqrt{\cosh^2(y)+\sinh^4(y)}\cosh(y)}$$ Now, note that \(\cosh^2(y) - \sinh^2(y) = 1\), which allows us to further simplify the integral: $$\int \frac{\cosh(y)}{\sqrt{\cosh^2(y)+\sinh^4(y)}} \frac{dx}{\cosh(y)}$$ We can eliminate the \(\cosh(y)\) from the numerator and denominator: $$\int \frac{dx}{\sqrt{\cosh^2(y)+\sinh^4(y)}}$$ #tag_title# Step 4: Evaluate the integral #tag_content# Replace \(x\) back in terms of \(y\): $$\int \frac{dx}{\sqrt{\cosh^2(y)+(\sinh^2(y))^2}}$$ Since there is no known closed form for this integral, we can use numerical methods to evaluate it. The final answer will depend on the given bounds of integration.

Step-by-step solution

Define 'u' for substitution

Let's perform a substitution by defining a new variable \(u\). Since the expression inside the square root is quite complicated, we should choose that to be our \(u\). Let: $$u = 1 + x^2 + x^4$$ Now, differentiate 'u' with respect to 'x': $$\frac{du}{dx} = 0 + 2x + 4x^3$$

Rearrange the du/dx equation for dx

Now, let's solve for \(dx\): $$dx = \frac{du}{2x + 4x^3}$$

Replace dx and expression within square root

Now substitute the expressions for \(u\) and \(dx\) into the integral: $$\int \frac{(1+x^2)}{(1-x^2)} \frac{du}{\sqrt{u}(2x + 4x^3)}$$

Simplify the integral

We can now simplify the whole integral. Notice that: $$\frac{(1+x^2)}{(1-x^2)} = \frac{(2x^2 - x^2 + 1)}{(1 - x^2)}$$ Now substitute this back into the integral: $$\int \frac{(2x^2 - x^2 + 1)}{(1 - x^2)} \frac{du}{\sqrt{u}(2x + 4x^3)}$$ As we can see, the integral is still quite complex.

Use sec substitution

Maybe we can simplify this further. Let's try using the trigonometric substitution \(x = \tanh(t)\): Rewriting the integral with \(x = \tanh(t)\): $$\int \frac{(2\left(\tanh^2(t)\right) - \left(\tanh^2(t)\right) + 1)}{(1 - \left(\tanh^2(t)\right))} \frac{du}{\sqrt{u}(2\tanh(t) + 4(\tanh^3(t)))}$$ Now we know that $$1 - \tanh^2(t) = \operatorname{sech}^2(t)$$ So the integral becomes: $$\int \frac{(2\left(\tanh^2(t)\right) - \left(\tanh^2(t)\right) + 1)}{(\operatorname{sech}^2(t))} \frac{du}{\sqrt{u}(2\tanh(t) + 4(\tanh^3(t)))}$$ The integral looks quite complicated now. We will have to re-analyze the problem and find a more suitable substitution or method of approach. This part of the solution is provided for complete transparency. RouteServiceProvider

Key concepts

U-Substitution

In integral calculus, the technique of u-substitution is a method used to simplify integrals by transforming a complex integral into a simpler one through a change of variables. The essence of this technique is to choose a part of the integral's integrand to be a new variable, usually denoted as u, and express the integral in terms of this new variable.

Let's consider our integral exercise. The denominator within the square root, 1+x^2+x^4, is a candidate for substitution since it can potentially simplify the integral. By choosing u = 1 + x^2 + x^4, it allows the integration to be written in terms of u which, as the next steps will show, can streamline the integration process. Remember to also derive du/dx to express dx in terms of du. This is crucial as it changes the differential element from dx to du.

However, the success of the u-substitution method significantly depends on the correct identification of the u part and its derivative as well as how they fit back into the integral to simplify it. The method is particularly powerful when the derivative of u, du/dx, is present elsewhere in the integrand or can be cancelled out when substituted back into the integral.

Trigonometric Substitution

Another useful technique in integral calculus is trigonometric substitution. This method involves substituting trigonometric functions for expressions to reshape the integral into a form that is easier to evaluate. Trigonometric substitution often comes in handy when dealing with square roots or quadratics.

In our problem, after applying u-substitution, the integral remains complex. Attempting trigonometric substitution by setting x = tanh(t) seems promising because hyperbolic trigonometric identities may simplify the expression, especially those involving sech and tanh. Recall that 1 - tanh^2(t) = sech^2(t), showing that these identities can transform the integral into trigonometric terms which can often be simpler to integrate. Despite this, if the resulting integral still appears complex or unwieldy, this may indicate that an alternative approach or a different substitution should be considered. It is important when using trigonometric substitution to be familiar with trigonometric identities and properties as they are instrumental in simplifying the resultant expression.

Integral Simplification

Reducing an integral to its simplest form, known as integral simplification, can make it much more approachable to evaluate. The goal is to break down or transform the integral so that standard integration methods can be applied with ease. This often involves algebraic manipulation, trigonometric identities, or substitution methods.

In the integral from our example, even after applying u-substitution and considering trigonometric substitution with hyperbolic functions, the result did not lead us to a straightforward path to the solution. At this point, it's vital to reassess the chosen strategies. The key to integral simplification is to iterate through different potential methods and substitutions, observing how they transform the integral. Frequently, splitting the integrand into smaller, more manageable parts, or factoring expressions is another path to simplification that could present a more direct route to finding the antiderivative. Ultimately, it is sometimes the case that what initially appears to simplify may still lead to complexity, and backtracking to explore other approaches is a normal part of solving complex integrals.