Question

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $$ \int \frac{d x}{x^{2} \sqrt{x^{2}+1}} $$

Short answer

The integral is \(-\frac{\sqrt{x^2 + 1}}{x} + C\).

Step-by-step solution

Choose the Trigonometric Substitution

For integrals involving expressions like \( \sqrt{x^2 + a^2} \), we use the substitution \( x = a \tan(\theta) \). Here, \( a = 1 \), so we choose \( x = \tan(\theta) \). This implies that \( dx = \sec^2(\theta) \, d\theta \).

Substitute in the Integral

Substitute \( x = \tan(\theta) \) and \( dx = \sec^2 (\theta) \, d\theta \) into the integral:\[\int \frac{d x}{x^2 \sqrt{x^2 + 1}} = \int \frac{\sec^2 (\theta) \, d\theta}{\tan^2 (\theta) \sqrt{\tan^2 (\theta) + 1}}\]

Simplify the Integral

Inside the square root, \( \tan^2 (\theta) + 1 = \sec^2 (\theta) \). Therefore, the integral becomes:\[\int \frac{\sec^2 (\theta) \, d\theta}{\tan^2 (\theta) \cdot \sec(\theta)} = \int \frac{\sec(\theta) \, d\theta}{\tan^2 (\theta)}\]

Express in Terms of Sine and Cosine

Recall that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Substitute these into the integral:\[\int \frac{\frac{1}{\cos(\theta)} \, d\theta}{\left( \frac{\sin(\theta)}{\cos(\theta)} \right)^2} = \int \frac{\cos(\theta) \, d\theta}{\sin^2(\theta)}\]

Apply Substitution

Let \( u = \sin(\theta) \), then \( du = \cos(\theta) \, d\theta \). Therefore, the integral becomes:\[\int \frac{\cos(\theta) \, d\theta}{\sin^2(\theta)} = \int \frac{1}{u^2} \cdot du = \int u^{-2} du\]

Integrate

Integrate \( u^{-2} \), which gives:\[\int u^{-2} du = -u^{-1} + C = -\frac{1}{u} + C\]

Substitute Back

Substitute \( u = \sin(\theta) \) back into the expression:\[-\frac{1}{\sin(\theta)} + C = -\csc(\theta) + C\]Recall from the substitution \( x = \tan(\theta) \), we have \( \tan(\theta) = x \), so \( \theta = \tan^{-1}(x) \) and \( \sin(\theta) = \frac{x}{\sqrt{x^2 + 1}} \). The expression becomes:\[-\frac{\sqrt{x^2 + 1}}{x} + C\]

Final Expression in Original Variable

Thus, the integrated result in terms of \( x \) is:\[\int \frac{dx}{x^2 \sqrt{x^2 + 1}} = -\frac{\sqrt{x^2 + 1}}{x} + C\]

Key concepts

Integration Techniques

Integration techniques are methods used in calculus to find the integral of functions. When faced with complex integrals, like the one in our exercise, \( \int \frac{d x}{x^{2} \sqrt{x^{2}+1}} \), one efficient approach is to use substitution methods such as trigonometric substitution.

In this integration technique, we use trigonometric identities to simplify expressions involving square roots. For expressions of the form \( \sqrt{x^2 + a^2} \), substituting \( x = a \tan(\theta) \) can often simplify the integral. Once we make the substitution, we replace \( dx \) with \( \sec^2(\theta) \, d\theta \) based on the derivative of the trigonometric function.

This substitution transforms the integral into a function of \( \theta \), where trigonometric identities can further simplify the expression, often converting complicated algebraic expressions into manageable integrals of basic trigonometric or algebraic forms.

Calculus Problem Solving

Solving calculus problems involves breaking down a problem into smaller, manageable parts and applying appropriate integration techniques, like the trigonometric substitution method shown in the original solution.

The key steps in solving the problem begin with recognizing patterns in the integrand that resemble forms involving \( \sqrt{x^2 + a^2} \). Upon recognizing this form, use substitution to reframe the integral in terms of simpler, trig-based expressions.

Next, carefully substitute in all the necessary trigonometric identities to reduce the complexity of the equation. In our example, rewriting \( \tan^2(\theta) + 1 \) as \( \sec^2(\theta) \) dramatically simplifies the task. After simplification, re-express the integral using basic trigonometric functions and perform another substitution if necessary, as we did with \( u = \sin(\theta) \).

Finally, after integrating, it's essential to revert all the substitutions to return to the original variable. Patience and attention to detail in each step will ensure that you correctly solve these calculus problems.

Advanced Integration Methods

Advanced integration methods often involve techniques such as trigonometric substitution, integration by parts, or partial fractions.
Trigonometric substitution, in particular, is very effective for integrals containing square roots, which are prevalent in higher-level calculus problems.

The exercise uses advanced integration through trigonometric substitution, which converts a problematic algebraic integral into an easier one involving trigonometric functions. Key to this is understanding and using trigonometric identities and derivatives. Recognizing when to apply these substitutions can simplify the integration process, allowing for the resolution of otherwise difficult integrals.

In this specific example, after conducting the trigonometric substitution and simplifying the integral, one performs a second substitution that makes the remaining integral a straightforward power function, which is easy to integrate. This method showcases the power of advanced integration techniques in simplifying and solving complex integrals.