Question

Evaluate the following integrals. $$\int \frac{x}{\sqrt{1-9 x^{2}}} d x$$

Short answer

After using the substitution method and simplifying the integral, we find that the integral of the given function, $$\int \frac{x}{\sqrt{1-9 x^{2}}} d x,$$ is equal to $$-\frac{1}{9}\sqrt{1-9x^2} + C,$$ where C is the constant of integration.

Step-by-step solution

Choose a substitution

First, let's choose a suitable substitution. Here, we can see that the denominator can be written in the form of a squared trigonometric function if we rewrite it as follows: $$1-9 x^{2} = 1 - (3x)^2.$$ Notice that this expression is similar to the Pythagorean Identity \(1 - \sin^2(\theta) = \cos^2(\theta)\). So, let's choose the substitution: $$3x = \sin(\theta).$$

Substitute and find the differential

Now that we have chosen the substitution \(3x = \sin(\theta)\), we need to find the differential, which is \(d(3x) = 3dx\). Differentiating \(3x = \sin(\theta)\) with respect to \(\theta\): $$3 \frac{d x}{d \theta} = \cos(\theta) \implies d x = \frac{\cos(\theta)}{3} d \theta.$$ Substitute the original integral with the given substitution and the differential: $$\int \frac{x}{\sqrt{1-9 x^{2}}} d x = \int \frac{\frac{\sin(\theta)}{3}}{\sqrt{1-\sin^2(\theta)}} \cdot \frac{cos(\theta)}{3} d \theta.$$

Simplify the integral

We can now simplify the integral expression by noticing that \(1 - \sin^2(\theta) = \cos^2(\theta)\): $$\int \frac{\sin(\theta)}{9 \sqrt{\cos^2(\theta)}} \cdot \cos(\theta) d \theta = \int \frac{\sin(\theta) \cos(\theta)}{9 \cos(\theta)} d \theta =\frac{1}{9} \int \sin(\theta) d \theta.$$

Evaluate the integral

Now, evaluate the integral of \(\sin(\theta)\): $$\frac{1}{9} \int \sin(\theta) d \theta = \frac{-1}{9}\cos(\theta) + C,$$ where C is the constant of integration.

Back-substitute the original variable

To rewrite the result in terms of the original variable x, back-substitute \(\theta\) using the inverse of our original substitution: $$x = \frac{\sin(\theta)}{3} \implies \theta = \arcsin(3x).$$ Now, substitute this back into our result: $$-\frac{1}{9}\cos(\theta) + C = -\frac{1}{9}\cos(\arcsin(3x)) + C.$$ Finally, using the Pythagorean Identity and the properties of inverse trigonometric functions, compute the cosine of \(\arcsin(3x)\): $$\cos(\arcsin(3x)) = \sqrt{1 - \sin^2(\arcsin(3x))} = \sqrt{1 - (3x)^2} = \sqrt{1-9x^2}.$$ Thus, the final result is: $$-\frac{1}{9}\cos(\arcsin(3x)) + C = -\frac{1}{9}\sqrt{1-9x^2} + C.$$

Key concepts

Substitution Method

The substitution method is a powerful technique in calculus used to simplify the integration process by changing the variable of integration. This method is particularly effective when dealing with complex expressions, allowing us to transform them into simpler forms.
To apply the substitution method, follow these steps:

• Select an appropriate substitution: Identify a part of the integral that can be replaced with a single variable. This often involves recognizing patterns that match known formulas or simplifications.
• Replace the variable: Substitute the chosen expression with a new variable, and adjust the differential accordingly. This means you need to express the entire integral in terms of this new variable and its corresponding differential.
• Simplify and integrate: The substitution should simplify the original integral into something more manageable. If done correctly, you'll have a new integral that is easier to solve.

Substitution helps transform the problem, allowing for straightforward integration. A clever choice of substitution is often key to solving integrals efficiently.

Trigonometric Substitution

Trigonometric substitution is an extended application of the substitution method specifically useful for integrals involving expressions like \(\sqrt{a^2 - x^2}\). It leverages trigonometric identities to vastly simplify these expressions.
Here's how it works:

• Identify suitable trigonometric identities: Each standard form of radical has a corresponding trigonometric function that simplifies it. For example, \(1 - x^2\) can be turned into \(\cos^2(\theta)\) using the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\).
• Select a trigonometric substitution: Replace the variable \(x\) with a trigonometric function. For instance, in \(\sqrt{1 - x^2}\), using \(x = \sin(\theta)\) transforms the expression into \(\cos^2(\theta)\).
• Solve resulting trigonometric integrals: The original integral transforms into one involving a trigonometric function, which is often simpler to handle.
• Translate back to the original variable: After evaluating the integral, use inverse trigonometric functions to switch back to the original variable.

Trigonometric substitution can make difficult integrals more accessible by utilizing the properties of trigonometric functions and their relationships.

Inverse Trigonometric Functions

Inverse trigonometric functions are vital in calculus, especially for understanding and expressing angles in terms of standard trigonometric functions like sine, cosine, and tangent.
These functions include:

• Arcsine \(\arcsin\): This function is the inverse of the sine function. If \(y = \sin(x)\), then \(x = \arcsin(y)\). It is used to find angles whose sine is a given number.
• Arccosine \(\arccos\): As the inverse of the cosine function, it provides the angle whose cosine is the number in question.
• Arctangent \(\arctan\): This function inverses the tangent function to find an angle with a specific tangent value.

Inverse trigonometric functions are essential when switching back to the original variable after solving an integral with trigonometric substitution. They allow conversions from a trigonometric expression back to algebraic ones.
These functions simplify expressing solutions in terms of initial variables, making them indispensable in calculus, especially when dealing with integrals and solving trigonometric equations.