Question

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $$ \int \frac{d x}{\left(1+x^{2}\right)^{3 / 2}} $$

Short answer

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

Step-by-step solution

Identify the Trigonometric Substitution

The integrand \( \frac{1}{(1+x^2)^{3/2}} \) suggests the substitution \( x = \tan(\theta) \) because \( 1 + x^2 \) becomes \( \sec^2(\theta) \) under this substitution. This simplifies the square root expression. Set \( x = \tan(\theta) \) so that \( dx = \sec^2(\theta) d\theta \).

Substitute and Simplify

Substitute \( x = \tan(\theta) \) and \( dx = \sec^2(\theta) d\theta \) into the integral: \[ \int \frac{dx}{(1+x^2)^{3/2}} = \int \frac{\sec^2(\theta) d\theta}{(\sec^2(\theta))^{3/2}}. \]This simplifies to:\[ \int \frac{\sec^2(\theta) d\theta}{\sec^3(\theta)} = \int \cos(\theta) d\theta. \]

Integrate with Respect to \( \theta \)

The integral \( \int \cos(\theta) d\theta \) is straightforward. It evaluates to:\[ \sin(\theta) + C, \]where \( C \) is the constant of integration.

Back-Substitute to Original Variable

Use the original substitution \( x = \tan(\theta) \) to find \( \sin(\theta) \). Recall from the substitution:\[ \sin(\theta) = \frac{x}{\sqrt{1+x^2}}. \]Therefore, the solution in terms of \( x \) is:\[ \frac{x}{\sqrt{1+x^2}} + C. \]

Key concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. One powerful approach is trigonometric substitution, especially useful for integrals involving square roots of quadratic expressions.
This technique transforms complex algebraic expressions into simpler trigonometric functions, making them easier to integrate.

• In the given problem, trigonometric substitution is used to simplify the integral \( \int \frac{d x}{(1+x^2)^{3/2}} \).
• The substitution \( x = \tan(\theta) \) helps in converting \( 1 + x^2 \) into \( \sec^2(\theta) \), as \( 1 + \tan^2(\theta) = \sec^2(\theta) \).
• This reduces the integrand to \( \int \cos(\theta) d\theta \), which is much more straightforward to integrate.

Mastering trigonometric substitution can simplify the process of solving challenging integrals by transforming them into more manageable forms.

Trigonometric Identities

Trigonometric identities provide the relationships between different trigonometric functions. They are extremely helpful in simplifying integrals that involve trigonometric functions.
In the domain of integration, these identities are indispensable.

• The essential identity involved in this problem is \( 1 + \tan^2(\theta) = \sec^2(\theta) \), which allows the substitution \( x = \tan(\theta) \) to transform the expression \( 1+x^2 \).
• After substitution, it leads to \( \int \cos(\theta) d\theta \), using the relationship \( \sec^3(\theta) = \sec^2(\theta)\cdot\sec(\theta) \).
• Recognizing these identities helps in the manipulation and simplification of integrals to arrive at a solvable form.

Knowing these trigonometric identities well makes the use of trigonometric substitution more effective and accessible.

Definite and Indefinite Integrals

In calculus, integrals can be classified as either definite or indefinite. An indefinite integral refers to the antiderivative of a function and includes a constant of integration \( C \).
A definite integral, on the other hand, computes the net area under a curve between two specific points.

• In this exercise, we're dealing with an indefinite integral, \( \int \frac{d x}{(1+x^2)^{3/2}} \), which means we're looking for a general form of the antiderivative.
• The indefinite integral is solved by performing the trigonometric substitution and then integrating with respect to \( \theta \).
• Finally, back-substitute \( \sin(\theta) \) as \( \frac{x}{\sqrt{1+x^2}} \) to express the result in terms of the original variable \( x \).

Understanding the nature of indefinite integrals allows us to transform them and find versatile solutions applicable to various conditions.