Question

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

Short answer

\( \frac{1}{2}x\sqrt{1+x^2} - \frac{1}{2}\ln|x+\sqrt{1+x^2}| + C \)

Step-by-step solution

Identify the Substitution

To integrate using trigonometric substitution, identify a substitution for the variable that will simplify the integral. In this case, we look at the expression \( \sqrt{1+x^{2}} \) and choose the substitution \( x = \tan(\theta) \). This will simplify the integral.

Compute the Differential

After choosing the substitution \( x = \tan(\theta) \), compute the differential \( dx \). Since \( x = \tan(\theta) \), we have \( dx = \sec^{2}(\theta) d\theta \).

Substitute into the Integral

Substitute \( x = \tan(\theta) \) and \( dx = \sec^{2}(\theta) d\theta \) into the integral. The integral becomes \( \int \frac{\tan^{2}(\theta) \sec^{2}(\theta) d\theta}{\sec(\theta)} = \int \tan^{2}(\theta) \sec(\theta) d\theta \).

Simplify the Integral with Trigonometric Identities

Use the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \) to simplify the integral: \( \int (\sec^2(\theta) - 1) \sec(\theta) d\theta \). This simplifies to \( \int (\sec^3(\theta) - \sec(\theta)) d\theta \).

Integrate Each Part

Integrate each part separately:- For \( \int \sec^3(\theta) d\theta \), use the reduction formula: \( \int \sec^3(\theta) d\theta = \frac{1}{2}(\sec(\theta)\tan(\theta) + \ln|\sec(\theta) + \tan(\theta)|) \).- Integrate \( \int \sec(\theta) d\theta = \ln|\sec(\theta) + \tan(\theta)| \).Combine these results to obtain the solution: \( \frac{1}{2}\sec(\theta)\tan(\theta) + \frac{1}{2}\ln|\sec(\theta) + \tan(\theta)| - \ln|\sec(\theta) + \tan(\theta)| \).

Simplify the Result

Factor and combine the logarithmic terms: \( \frac{1}{2}\sec(\theta)\tan(\theta) - \frac{1}{2}\ln|\sec(\theta) + \tan(\theta)| \).

Convert Back to x

Convert back to the original variable \( x \) using \( x = \tan(\theta) \) which gives \( \sec(\theta) = \sqrt{1 + x^2} \) and \( \tan(\theta) = x \). The final answer in terms of \( x \) is: \[ \frac{1}{2}x\sqrt{1+x^2} - \frac{1}{2}\ln|x+\sqrt{1+x^2}| + C \], where \( C \) is the constant of integration.

Key concepts

Integration Techniques

When approaching calculus problems, particularly integration, there are several methods at your disposal to simplify and solve the problem. For integrals involving square roots and quadratic terms, the technique of trigonometric substitution proves extremely helpful. This method uses the fact that trigonometric functions can transform complicated algebraic forms into simpler, integrable expressions.

In our example, we were dealing with the integral \[ \int \frac{x^{2} dx}{\sqrt{1+x^{2}}}\] The trigonometric substitution here is to let \( x = \tan(\theta) \), transforming the integral to involve trigonometric rather than algebraic functions. This choice is strategic because the identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \) simplifies the square root component of the integral. After computing the differential \( dx = \sec^2(\theta) d\theta \), the complex algebraic expression becomes much simpler to handle.The main steps in this technique involve:

• Identifying a substitution that will simplify the integral based on known trigonometric identities.
• Changing the variable and the differential to trigonometric forms.
• Solving the resulting trigonometric integral and then converting back to the original variable.

Trigonometric Identities

Trigonometric identities are mathematical relations that describe certain inherent relationships between different trigonometric functions, making them essential tools for simplifying integrals in calculus problems. In particular, these identities help in transforming and simplifying expressions that may seem difficult at first glance into more manageable forms.

For instance, the identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \) was pivotal in our simplification process. By substituting \( x = \tan(\theta) \), the expression under the square root, \( \sqrt{1 + x^2} \), becomes \( \sqrt{\sec^2(\theta)} = \sec(\theta) \). This transformation helps turn the integral into a simpler form: \( \int \tan^2(\theta) \sec(\theta) d\theta \), which can be further simplified using the identity mentioned.

Furthermore, other identities such as \( \sec^2(\theta) = 1 + \tan^2(\theta) \) helped in calculating differentials. Understanding these relationships allows the integral to be solved step by step, replacing complex algebraic terms with trigonometric ones which are more straightforward to integrate.

Calculus Problems

Calculus problems often involve finding areas under curves, slopes of tangents, or solving limits, and integration forms a core part of it. Integrals like \( \int \frac{x^{2} dx}{\sqrt{1+x^{2}}} \) utilize calculus to find solutions where algebra alone might struggle. With trigonometric substitution, calculus provides a pathway through challenging integrals by using clever substitutions and simplifying the problem into a more manageable form.

One critical aspect of solving calculus problems is interpreting the final result back in terms of the original variable. Once the integration is completed in the world of trigonometric variables, it's necessary to revert back to the original variables, such as converting \( \theta \) back to \( x \). Using the relationships from the substitution, such as \( x = \tan(\theta) \) and \( \sec(\theta) = \sqrt{1 + x^2} \), helps in expressing the final answer in a form recognizable in the context of the problem.

The robustness of calculus and its techniques offer powerful tools for tackling a wide range of mathematics problems. Integrals, derivatives, and limits form a foundation upon which more complex mathematics is built, making them essential topics to master for anyone studying math or related fields.