Question

The following integration formulas yield inverse trigonometric functions: \(1 .\) $$ \int \frac{d u}{\sqrt{a^{2}-u^{2}}}=\sin ^{-1} \frac{u}{a}+C $$ \(2 .\) $$ \int \frac{d u}{a^{2}+u^{2}}=\frac{1}{a} \tan ^{-1} \frac{u}{a}+C $$ \(3 .\) $$ \int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \sec ^{-1} \frac{u}{a}+C $$

Short answer

These formulas provide the solutions to integrals resulting in inverse trigonometric functions: arcsin, arctan, and arcsec.

Step-by-step solution

Identify the Formulas

We'll first identify the integral forms and their respective solutions. These integrals are associated with the inverse trigonometric functions: arcsin, arctan, and arcsec respectively.

Integration 1: Using Arcsin

The integral \( \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1} \frac{u}{a} + C \) is used when the integrand matches the form \( \frac{1}{\sqrt{a^2 - u^2}} \). This integral results in the inverse sine function (\( \sin^{-1} \)).

Integration 2: Using Arctan

For \( \int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1} \frac{u}{a} + C \), it is used when the integrand matches the form \( \frac{1}{a^2 + u^2} \). This integral results in the inverse tangent function (\( \tan^{-1} \)).

Integration 3: Using Arcsec

The integral formula \( \int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} \sec^{-1} \frac{u}{a} + C \) is applied when the integrand matches the form \( \frac{1}{u \sqrt{u^2 - a^2}} \). This computation yields the inverse secant function (\( \sec^{-1} \)).

Key concepts

Integration Formulas

The world of calculus unfolds some fascinating techniques, and integration is one of its core pillars. When you come across intricate expressions, certain integration formulas allow us to find solutions utilizing inverse trigonometric functions. These functions can simplify the process of finding antiderivatives.
Integration formulas refer to standard methods used to integrate across different types of functions, often leading to specific well-known results. In the integration world, inverse trigonometric functions frequently appear, acting as solutions to integral expressions.

• Arcsin formula: This involves expressions of the form \( \int \frac{du}{\sqrt{a^2 - u^2}} \), resulting in an inverse sine function, \( \sin^{-1} \frac{u}{a} + C \).
• Arctan formula: Used for expressions \( \int \frac{du}{a^2 + u^2} \). It leads to an inverse tangent function, \( \frac{1}{a} \tan^{-1} \frac{u}{a} + C \).
• Arcsec formula: For the form \( \int \frac{du}{u \sqrt{u^2 - a^2}} \), the result is an inverse secant function, \( \frac{1}{a} \sec^{-1} \frac{u}{a} + C \).

These integrals help us evaluate expressions that naturally align with geometric interpretations often linked to circular or angular concepts.

Arcsin Function

Inverse trigonometric functions are like gateways between algebraic expressions and geometric interpretations. The arcsin function, also known as the inverse sine, is one such gateway. Represented as \( \sin^{-1}(x) \), it reverses the sine function.
When dealing with integration, the arcsin function comes in handy to evaluate integrals with the form \( \int \frac{du}{\sqrt{a^2 - u^2}} \). This integral appears commonly when you have expressions curving beneath a certain bound, metaphorically speaking.
What makes arcsin special is its range:

• The domain of \( \sin^{-1}(x) \) is \(-1 \leq x \leq 1\), making it applicable where the sine function's results reside.
• The range of the function is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), meaning the solutions reside in the first and fourth quadrants of the unit circle.

The arcsin function holds diverse applications in physics and engineering, particularly in problems involving wave heights, angles of elevation, or even sound waves.

Arctan Function

The arctan function, or inverse tangent \( \tan^{-1}(x) \), is vital for converting tangent-related equations back to angles. Imagine you have an angle whose tangent you know; arctan helps you find the angle itself.
In integration, the arctan function is useful to solve integrals in the form of \( \int \frac{du}{a^2 + u^2} \). This setup commonly represents situations where horizontal asymptotes exist, or integer slopes are involved.

• Domain: While the tangent function spans all real numbers, arctan is clearly defined for all real values of \(x\).
• Range: The range of \( \tan^{-1}(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which means the solution falls within the open interval of negative and positive 90 degrees.

Arctan is commonly used in contexts like waves, signal processing, or even in navigation systems where angles and gradients need explicit computations.

Arcsec Function

The arcsecant function, represented by \( \sec^{-1}(x) \), helps to decode angles from their secant values. Given the understanding that secant is the reciprocal of cosine, arcsec is its inverse.
When integrating, you often stumble upon expressions \( \int \frac{du}{u \sqrt{u^2 - a^2}} \) that find their solutions through the arcsec function. This is common with scenarios requiring the evaluation of hyperbolic paths or segments.

• Domain: The function \( \sec^{-1}(x) \) restricts input values to intervals \( |x| \geq 1 \) because secant includes only these values.
• Range: The possible outputs fall within the intervals \( \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \), capturing positive and negative angles beyond the first quadrant.

Arcsec is particularly significant in disciplines requiring computations of distances, optics, or where circular/elliptical phenomena exist.