Question

Find the antiderivative of \(\int \frac{d x}{\sqrt{1-16 x^{2}}}\).

Short answer

The antiderivative is \( \frac{1}{4} \sin^{-1}(4x) + C \).

Step-by-step solution

Identify the Form of the Integral

The given integral is \( \int \frac{dx}{\sqrt{1-16x^{2}}} \). This integral resembles the standard form of inverse trigonometric functions. In particular, it is similar to the derivative of the inverse sine function.

Recall the Standard Integral Formula

The standard formula for the antiderivative of \( \frac{1}{\sqrt{1 - u^2}} \) is \( \sin^{-1}(u) + C \), where \( C \) is the constant of integration.

Substitute and Simplify

We need to manipulate the current expression to fit the standard form \( \frac{1}{\sqrt{1-u^2}} \). Here, set \( u = 4x \). Consequently, \( du = 4dx \), or \( dx = \frac{1}{4} du \). Therefore, substitute \( x = \frac{1}{4} u \) into the integral: \[\int \frac{1}{4} \cdot \frac{1}{\sqrt{1 - u^2}} \, du \] This simplifies the expression to \( \frac{1}{4} \int \frac{1}{\sqrt{1-u^2}} \, du \).

Integrate using the Standard Formula

Now apply the formula from Step 2: \[\frac{1}{4} \int \frac{1}{\sqrt{1 - u^2}} \, du = \frac{1}{4} \sin^{-1}(u) + C\] Substitute back \( u = 4x \): \[= \frac{1}{4} \sin^{-1}(4x) + C\]

Write the Final Antiderivative

Therefore, the antiderivative of the original function is given by \[\int \frac{dx}{\sqrt{1-16x^{2}}} = \frac{1}{4} \sin^{-1}(4x) + C\] Where \( C \) is the constant of integration.

Key concepts

Inverse Trigonometric Functions

Inverse Trigonometric Functions are mathematical functions that reverse the effect of the usual trigonometric functions like sine, cosine, and tangent. In the context of integration, they are often used to solve integrals that involve expressions under a square root resembling a trigonometric identity.
For instance, the integral \( \int \frac{dx}{\sqrt{1-16x^{2}}} \) is identified as similar to the form associated with the inverse sine function. The antiderivative involving \( \frac{1}{\sqrt{1-u^2}} \) gives rise to the function \( \sin^{-1}(u) \), where "inverse sine" or \( \sin^{-1} \) denotes the function that satisfies \( \sin(\sin^{-1}(x)) = x \) for \( -1 \leq x \leq 1 \).

• Applications of inverse trigonometric functions are crucial in calculus for finding antiderivatives or solving integration problems, especially those that model cyclic or wave-based phenomena.
• Understanding the relation between geometrical interpretation (such as angle measures) and these functions enhances their application in solving real-world problems.

Integration Techniques

Integration Techniques are various methods employed to compute the antiderivative or the area under a curve for given functions. These techniques can range from basic rules of integration to more advanced methods designed to solve complex integrals.
When encountering an integral such as \( \int \frac{dx}{\sqrt{1-16x^{2}}} \), it is imperative to recognize its form and compare it to known integral formulas. This recognition facilitates the application of specific techniques, such as substituting variables to fit a standard antiderivative formula.
Simplifying integrals by adjusting constants or terms within the equation can often make integration more approachable. Here, identifying the matching form of \( \frac{1}{\sqrt{1-u^2}} \) shows us the direct application of the arc sine (or \( \sin^{-1} \)) formula is required. This makes it crucial to:

• Recognize standard forms and related formulas.
• Reapply known standard solutions effectively.
• Understand the procedure behind these techniques to enrich problem-solving skills.

Substitution Method

The Substitution Method, also called \( u \)-substitution, is a useful integration technique that simplifies the process by transforming the given variable of integration into another variable. This method usually turns a complicated function into a simpler one by changing the variable.
Applying substitution in \( \int \frac{dx}{\sqrt{1-16x^{2}}} \), we set \( u = 4x \). This changes the differential with \( du = 4dx \), allowing us to reframe the integral in terms of \( u \). When substituted, the integral becomes \( \frac{1}{4} \int \frac{1}{\sqrt{1-u^{2}}} \, du \).
Substitution allows us to use our knowledge of the \( \sin^{-1} \) function, simplifying the integral into a standard form. Thus, the substitution method steps are as follows:

• Select a suitable substitution, \( u = \text{expression involving the original variable} \).
• Compute \( du \) and express \( dx \) in terms of \( du \).
• Convert the integral into the new variable \( u \) and solve.
• Substitute back to return to the original variable.

Using substitution effectively changes the perspective on a problem, making what seems difficult much easier to solve.