Problem 34
Question
Evaluate the indefinite integral. \(\int \frac{\sin x}{1+\cos ^{2} x} d x\)
Step-by-Step Solution
Verified Answer
\(-\tan^{-1}(\cos x) + C\)
1Step 1: Recognize the Form
The integrand is \( \frac{\sin x}{1+\cos^2 x} \). Notice that the denominator is a function of \( \cos x \), which suggests that substitution might simplify the integration.
2Step 2: Use Trigonometric Substitution
Let \( u = \cos x \). Then, the derivative \( du = -\sin x \, dx \) suggests that \( \sin x \, dx = -du \). Substitute these into the integral.
3Step 3: Simplify the Integral
By substituting, the integral becomes \( \int \frac{-du}{1+u^2} \). The negative sign can be factored out, giving \(- \int \frac{du}{1+u^2} \).
4Step 4: Identify Integral Formula
The integral \( \int \frac{du}{1+u^2} \) is a standard result, known to be \( \tan^{-1} u + C \), where \( C \) is the constant of integration.
5Step 5: Substitute Back
Replace \( u \) with \( \cos x \) in the integrated result: \(- (\tan^{-1}(\cos x) + C) = -\tan^{-1}(\cos x) + C\).
6Step 6: Write Final Form
The final form of the indefinite integral is \(-\tan^{-1}(\cos x) + C\).
Key Concepts
Trigonometric SubstitutionIndefinite IntegralIntegration Techniques
Trigonometric Substitution
Trigonometric substitution is a powerful technique used in calculus to simplify integrals involving trigonometric functions. In this exercise, we aim to simplify the integration process of the integral \( \int \frac{\sin x}{1+\cos^2 x} \, dx \). Trigonometric substitution helps to achieve this simplification by substituting one trigonometric function for another to make the integration easier.
The idea is to identify a substitution that will transform the integral into a more familiar form. In our example, we look at the denominator \(1 + \cos^2 x\) and choose \(u = \cos x\). This substitution changes the integral into a form where it becomes easier to integrate, because the differential equation becomes simpler with \(du = -\sin x\ dx\). This means that \( \sin x \, dx = -du\), replacing part of the integrand and simplifying the expression.
Such substitutions are particularly useful when dealing with integrals that involve squares of trigonometric functions, or a combination of these. Each substitution varies based on the form of the integral, but the goal is always to transform the original integral into one of a simpler form.
The idea is to identify a substitution that will transform the integral into a more familiar form. In our example, we look at the denominator \(1 + \cos^2 x\) and choose \(u = \cos x\). This substitution changes the integral into a form where it becomes easier to integrate, because the differential equation becomes simpler with \(du = -\sin x\ dx\). This means that \( \sin x \, dx = -du\), replacing part of the integrand and simplifying the expression.
Such substitutions are particularly useful when dealing with integrals that involve squares of trigonometric functions, or a combination of these. Each substitution varies based on the form of the integral, but the goal is always to transform the original integral into one of a simpler form.
Indefinite Integral
The concept of an indefinite integral is foundational in calculus, representing the accumulation of quantities and the area under curves. Unlike definite integrals, indefinite integrals do not have upper and lower limits of integration. Instead, they seek to find a general function, known as an antiderivative, whose derivative yields the original integrand.
In our example exercise, the indefinite integral \(\int \frac{\sin x}{1+\cos^2 x} \, dx\) is evaluated. The process involves finding a function whose derivative is \(\frac{\sin x}{1+\cos^2 x}\). By making an appropriate substitution, we simplified the integral into a standard form that we can easily integrate.
Indefinite integrals are always accompanied by a constant of integration, \(C\). This constant accounts for the family of functions with the same derivative but differing by a constant value. Therefore, the solution incorporates this constant, conveying the general solution to the problem.
In our example exercise, the indefinite integral \(\int \frac{\sin x}{1+\cos^2 x} \, dx\) is evaluated. The process involves finding a function whose derivative is \(\frac{\sin x}{1+\cos^2 x}\). By making an appropriate substitution, we simplified the integral into a standard form that we can easily integrate.
Indefinite integrals are always accompanied by a constant of integration, \(C\). This constant accounts for the family of functions with the same derivative but differing by a constant value. Therefore, the solution incorporates this constant, conveying the general solution to the problem.
Integration Techniques
Various integration techniques exist to solve complex integrals, making calculus a versatile and powerful mathematical tool. In this particular problem, we employed a technique called 'trigonometric substitution' which was well-suited to simplify the integration process.
Once the substitution \(u = \cos x\) was made, the integral transformed into \(- \int \frac{du}{1+u^2}\). Knowledge of integral formulas then became essential. Here, the integral \(\int \frac{du}{1+u^2}\) is recognized as the derivative of the inverse tangent function, namely \(\tan^{-1} u\). It’s a standard result used often in integration.
In addition to substitution, there are other techniques like integration by parts, partial fraction decomposition, and using tables of integrals. The choice of technique depends on the form of the integrand and often necessitates creativity and practice to determine which is the most efficient path to a solution. Each technique is a tool to simplify the problem at hand, and proficiency in these techniques is built with experience.
Once the substitution \(u = \cos x\) was made, the integral transformed into \(- \int \frac{du}{1+u^2}\). Knowledge of integral formulas then became essential. Here, the integral \(\int \frac{du}{1+u^2}\) is recognized as the derivative of the inverse tangent function, namely \(\tan^{-1} u\). It’s a standard result used often in integration.
In addition to substitution, there are other techniques like integration by parts, partial fraction decomposition, and using tables of integrals. The choice of technique depends on the form of the integrand and often necessitates creativity and practice to determine which is the most efficient path to a solution. Each technique is a tool to simplify the problem at hand, and proficiency in these techniques is built with experience.
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Problem 34
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