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1.7 Integrals Resulting in Inverse Trigonometric Functions

3 min read•june 24, 2024

Inverse trigonometric integrals are a key part of calculus, linking integration to familiar functions like sine and tangent. These integrals pop up in various fields, from physics to engineering, helping solve real-world problems involving circular and periodic motion.

Mastering these integrals requires recognizing specific forms and applying the right techniques. You'll learn to use substitution, handle , and interpret results geometrically. This knowledge will boost your problem-solving skills and deepen your understanding of calculus concepts.

Integrals Resulting in Inverse Trigonometric Functions

Inverse trigonometric function integrals

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  • Recognize common forms of integrals that yield as their solutions
    • ∫1a2−x2[dx](https://www.fiveableKeyTerm:dx)=sin⁡−1(xa)+C\int \frac{1}{\sqrt{a^2-x^2}} [dx](https://www.fiveableKeyTerm:dx) = \sin^{-1}(\frac{x}{a}) + C for ∣x∣<∣a∣|x| < |a| ()
    • ∫1a2+x2dx=1atan⁡−1(xa)+C\int \frac{1}{a^2+x^2} dx = \frac{1}{a}\tan^{-1}(\frac{x}{a}) + C for all real xx ()
    • ∫1x2−a2dx=ln⁡∣x+x2−a2∣+C\int \frac{1}{\sqrt{x^2-a^2}} dx = \ln|x+\sqrt{x^2-a^2}| + C for ∣x∣>∣a∣|x| > |a| (area of hyperbolic sector)
    • ∫1xx2−a2dx=1asec⁡−1∣xa∣+C\int \frac{1}{x\sqrt{x^2-a^2}} dx = \frac{1}{a}\sec^{-1}|\frac{x}{a}| + C for ∣x∣>∣a∣|x| > |a| (arcsecant)
  • Constant aa plays a crucial role in determining the domain restrictions and the argument of the resulting inverse trigonometric function
    • Value of aa affects the shape of the integrand and the feasible values for xx
    • Argument of the inverse function is typically a ratio involving xx and aa (xa\frac{x}{a})
  • Identify the appropriate integral formula to apply based on the structure of the given integrand
    • Match the integrand to one of the recognized forms to determine the corresponding inverse trigonometric solution
  • Understand the relationship between these integrals and the

Substitution for inverse trigonometric integrals

  • Employ trigonometric substitution to convert integrals into a form that results in an inverse trigonometric function
    • For a2−x2\sqrt{a^2-x^2}, substitute x=asin⁡θx = a\sin\theta and dx=acos⁡θdθdx = a\cos\theta d\theta
      • Transforms the integral into a trigonometric form that can be solved using an inverse trigonometric function
    • For x2+a2\sqrt{x^2+a^2}, substitute x=atan⁡θx = a\tan\theta and dx=asec⁡2θdθdx = a\sec^2\theta d\theta
      • Converts the integral into a form involving tan⁡θ\tan\theta and sec⁡θ\sec\theta, leading to an arctangent solution
    • For x2−a2\sqrt{x^2-a^2}, substitute x=asec⁡θx = a\sec\theta and dx=asec⁡θtan⁡θdθdx = a\sec\theta\tan\theta d\theta
      • Transforms the integral into a form involving sec⁡θ\sec\theta and tan⁡θ\tan\theta, resulting in an arcsecant or natural logarithm solution
  • Simplify the transformed integral using trigonometric identities and properties
    • Apply the appropriate inverse trigonometric integral formula to evaluate the simplified integral
  • Express the final result in terms of the original variable using the substitution relationship
    • Substitute back the original variable xx to obtain the solution in its original context
  • Consider using the when dealing with composite functions in these substitutions

Domain and geometry of inverse trigonometric integrals

  • Understand the domain restrictions associated with each inverse trigonometric integral formula
    • For sin⁡−1(xa)\sin^{-1}(\frac{x}{a}), xx must satisfy ∣x∣<∣a∣|x| < |a| (restricted domain)
    • For tan⁡−1(xa)\tan^{-1}(\frac{x}{a}), xx can take any real value (unrestricted domain)
    • For ln⁡∣x+x2−a2∣\ln|x+\sqrt{x^2-a^2}|, xx must satisfy ∣x∣>∣a∣|x| > |a| (restricted domain)
    • For sec⁡−1∣xa∣\sec^{-1}|\frac{x}{a}|, xx must satisfy ∣x∣>∣a∣|x| > |a| (restricted domain)
  • Interpret the geometric meaning of these integrals in terms of area or arc length
    • ∫1a2−x2dx\int \frac{1}{\sqrt{a^2-x^2}} dx represents the area of a circular sector with radius aa
      • Relates to the area of a portion of a circle bounded by an angle
    • ∫1a2+x2dx\int \frac{1}{a^2+x^2} dx represents the area under the curve y=1a2+x2y = \frac{1}{a^2+x^2}
      • Corresponds to the area between the curve and the xx-axis
    • ∫1x2−a2dx\int \frac{1}{\sqrt{x^2-a^2}} dx and ∫1xx2−a2dx\int \frac{1}{x\sqrt{x^2-a^2}} dx represent areas related to
      • Hyperbolic functions (sinh⁡\sinh, cosh⁡\cosh) are analogous to trigonometric functions but involve hyperbolas instead of circles

Advanced techniques for inverse trigonometric integrals

  • can be useful when dealing with products involving inverse trigonometric functions
  • decomposition may be necessary for more complex rational functions before applying inverse trigonometric integration
  • The is essential in understanding how these integrals relate to their antiderivatives

Key Terms to Review (22)

$ an^{-1}$: $ an^{-1}$ is the inverse trigonometric function of the tangent function. It represents the angle whose tangent is a given value. This term is particularly important in the context of integrals resulting in inverse trigonometric functions, as it allows for the evaluation of certain types of integrals involving tangent functions.
$ ext{sin}^{-1}$: $ ext{sin}^{-1}$ is the inverse sine function, which is used to find the angle whose sine is a given value. It represents the angle in the standard position of a unit circle that has a given sine value. This term is particularly relevant in the context of integrals resulting in inverse trigonometric functions, as the inverse sine function can arise in the solutions to certain types of integrals.
$ extsec^{-1}$: $ extsec^{-1}$ is the inverse of the secant function, which is one of the inverse trigonometric functions. It represents the angle whose secant is a given value. This term is particularly important in the context of integrals resulting in inverse trigonometric functions, as it allows for the evaluation of certain types of integrals involving the secant function. The inverse secant function, $ extsec^{-1}$, is a way to undo the secant function, just as the inverse sine, cosine, and tangent functions undo their respective trigonometric functions. It provides a means to find the angle given the secant value, which is useful in various applications, such as in the analysis of certain types of integrals.
Antiderivative: An antiderivative, also known as a primitive function or indefinite integral, is a function whose derivative is the original function. It represents the accumulation or the reverse process of differentiation, allowing us to find the function that was differentiated to obtain a given derivative.
Arcsine: The arcsine function, denoted as $\arcsin(x)$, is the inverse of the sine function. It is used to find the angle whose sine is equal to a given value.
Arctangent: Arctangent is the inverse function of the tangent function, which returns the angle whose tangent is a given number. It is often denoted as 'tan^{-1}(x)' or 'arctan(x)'. This function is crucial when dealing with integrals that involve trigonometric identities, particularly in evaluating areas under curves where the relationships between angles and their tangents come into play.
Chain Rule: The chain rule is a fundamental concept in calculus that allows for the differentiation of composite functions. It provides a systematic way to find the derivative of a function that is composed of other functions.
Derivative of Inverse Trigonometric Functions: The derivative of inverse trigonometric functions refers to the process of finding the rate of change or slope of the tangent line at a specific point on the graph of an inverse trigonometric function. This concept is particularly important in the context of integrals resulting in inverse trigonometric functions, as the derivative plays a crucial role in evaluating and simplifying these integrals.
Domain Restrictions: Domain restrictions refer to the limitations placed on the input values, or domain, of a function. This concept is particularly relevant when evaluating integrals that result in inverse trigonometric functions, as the domain of these functions is often restricted to ensure the integral has a unique and well-defined solution.
Dx: The term 'dx' represents an infinitesimally small change or increment in the independent variable 'x' within the context of integral calculus. It is a fundamental concept that connects the definite integral, the Fundamental Theorem of Calculus, integration formulas, inverse trigonometric functions, areas between curves, and various integration strategies.
Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus is a central result in calculus that establishes a deep connection between the concepts of differentiation and integration. It provides a powerful tool for evaluating definite integrals and understanding the relationship between the rate of change of a function and the function itself.
Hyperbolic functions: Hyperbolic functions are a set of mathematical functions that are analogs of the ordinary trigonometric functions but are based on hyperbolas instead of circles. They include hyperbolic sine ($$\sinh$$), hyperbolic cosine ($$\cosh$$), and others, which are essential in various calculus applications such as integrals, differential equations, and trigonometric substitution. These functions exhibit properties similar to trigonometric functions but have distinct geometric interpretations related to hyperbolas.
Indefinite integral: An indefinite integral represents the collection of all antiderivatives of a function, essentially reversing the process of differentiation. It is expressed in the form $$\int f(x) \, dx = F(x) + C$$, where $$F(x)$$ is the antiderivative of $$f(x)$$, and $$C$$ is a constant of integration that accounts for the fact that there are infinitely many antiderivatives differing only by a constant. Understanding indefinite integrals is crucial in various mathematical contexts, as they provide foundational techniques for solving equations and analyzing areas under curves.
Integration by Parts: Integration by parts is a technique used to integrate products of functions by transforming the integral into a simpler form using the formula $$\int u \, dv = uv - \int v \, du$$. This method connects various integration strategies, making it especially useful in situations where other techniques like substitution may not be effective.
Integration Techniques: Integration techniques refer to the various methods and strategies used to evaluate and solve integrals, which are fundamental operations in calculus. These techniques allow for the calculation of the area under a curve, the volume of a three-dimensional object, and the accumulation of quantities over a continuous domain.
Inverse Cosine: The inverse cosine, also known as the arccosine, is the inverse function of the cosine trigonometric function. It is used to find the angle whose cosine is a given value, allowing for the determination of the angle from the ratio of the adjacent and hypotenuse sides of a right triangle.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the standard trigonometric functions (sine, cosine, tangent, etc.). They allow us to find the angle given the value of a trigonometric function, which is essential in the context of integrals and integration by parts.
Partial Fractions: Partial fractions is a technique used to decompose a rational function into a sum of simpler rational functions. This method is often employed when integrating rational functions, as it allows for the use of inverse trigonometric functions, integration by parts, and other integration techniques.
Pythagorean identity: The Pythagorean identity is a fundamental relationship in trigonometry that states $$ ext{sin}^2(x) + ext{cos}^2(x) = 1$$ for any angle $$x$$. This identity connects the sine and cosine functions and is essential for solving various problems involving trigonometric functions, especially when working with integrals and substitutions that involve these functions.
Substitution Method: The substitution method is a technique used in calculus to evaluate integrals by replacing the original variable with a new variable that simplifies the integration process. This method is particularly useful when dealing with integrals that involve inverse trigonometric functions.
The Integral Symbol (∫): The integral symbol (∫) represents the mathematical operation of integration, which is the inverse of differentiation. It is used to calculate the accumulated change of a function over an interval, finding the area under a curve, or determining the total effect of a varying quantity.
U-Substitution Rule: The u-substitution rule is a fundamental technique in integral calculus that allows for the simplification of integrals by transforming the variable of integration. This method is particularly useful when dealing with integrals that involve composite functions, as it helps to isolate and evaluate the inner function more easily.
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