Problem 84
Question
Evaluate the integrals. $$\int \frac{\sqrt{\tan ^{-1} x} d x}{1+x^{2}}$$
Step-by-Step Solution
Verified Answer
\( \frac{2}{3} (\tan^{-1} x)^{3/2} + C \).
1Step 1: Identify Substitution
Notice that the integrand contains the function \( \sqrt{\tan^{-1} x} \). A suitable substitution is to let \( u = \tan^{-1} x \). This simplifies the integrand. We know that the derivative \( du = \frac{1}{1+x^2} dx \).
2Step 2: Substitute and Simplify
Using the substitution \( u = \tan^{-1} x \), we find \( du = \frac{1}{1+x^2} dx \). This setup allows us to rewrite the integral in terms of \( u \): \( \int \sqrt{u} \, du \).
3Step 3: Integrate with Respect to u
Integrate \( \int \sqrt{u} \, du \). Rewriting \( \sqrt{u} \) as \( u^{1/2} \), use the power rule for integration: \( \int u^{n} \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = \frac{1}{2} \), so the integral becomes \( \frac{u^{3/2}}{3/2} + C = \frac{2}{3} u^{3/2} + C \).
4Step 4: Substituting Back to x
Replace \( u \) with \( \tan^{-1} x \) to express the integral in terms of \( x \): \( \frac{2}{3} (\tan^{-1} x)^{3/2} + C \).
5Step 5: Final Answer
The evaluated integral is \( \frac{2}{3} (\tan^{-1} x)^{3/2} + C \), where \( C \) is the constant of integration.
Key Concepts
Substitution MethodTrigonometric IntegralsPower Rule for Integration
Substitution Method
The substitution method is a powerful technique in integral calculus that simplifies complex integrals by allowing us to replace a complicated part of the integrand with a simpler expression. The key idea is to change the variable of integration to make the integral easier to evaluate.
In our exercise, we identified that the function \( \sqrt{\tan^{-1} x} \) complicates the integral. By letting \( u = \tan^{-1} x \), we effectively turned the variable \( x \) into \( u \). Thus, the derivative \( du = \frac{1}{1+x^2} dx \) becomes a vital part of this substitution, helping us to transform the original integral into a simpler form.
Here’s what you do in the substitution method:
In our exercise, we identified that the function \( \sqrt{\tan^{-1} x} \) complicates the integral. By letting \( u = \tan^{-1} x \), we effectively turned the variable \( x \) into \( u \). Thus, the derivative \( du = \frac{1}{1+x^2} dx \) becomes a vital part of this substitution, helping us to transform the original integral into a simpler form.
Here’s what you do in the substitution method:
- Choose the substitution \( u = g(x) \) that simplifies the integral.
- Differentiate \( u \) to find \( du \).
- Replace \( dx \) and the function in terms of \( u \) and \( du \).
- Simplify the integral in the new variable.
- Evaluate and then substitute back the original variable to express your result in terms of \( x \).
Trigonometric Integrals
Trigonometric integrals are integrals that involve trigonometric functions. They are frequently encountered in calculus due to the periodic nature of these functions. These integrals often require specific strategies such as trigonometric identities or substitutions to solve.
In our problem, we've used the inverse tangent function, \( \tan^{-1} x \), a classic trigonometric function. Understanding this function's properties, including its derivative, is crucial when dealing with calculus problems involving trigonometry.
Here's a basic rundown when tackling trigonometric integrals:
In our problem, we've used the inverse tangent function, \( \tan^{-1} x \), a classic trigonometric function. Understanding this function's properties, including its derivative, is crucial when dealing with calculus problems involving trigonometry.
Here's a basic rundown when tackling trigonometric integrals:
- Identify the trigonometric function in your integrand.
- Consider appropriate substitutions, especially when inverse trigonometric functions are involved.
- Remember key derivatives such as \( \frac{d}{dx} [\tan^{-1} x] = \frac{1}{1+x^2} \).
- Use identities or substitutions to simplify the integral, making it easier to manage.
Power Rule for Integration
The power rule for integration is one of the fundamental techniques learned in integral calculus. It allows us to integrate simple power functions with ease and is applicable to functions of the form \( x^n \).
For any real number \( n eq -1 \), the rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
In our exercise, after applying substitution, the integral became \( \int \sqrt{u} \, du \), which can be rewritten as \( \int u^{1/2} \, du \). Here, the power rule shines. Setting \( n = \frac{1}{2} \), we integrated to get \( \frac{u^{3/2}}{3/2} + C \), simplifying it further to \( \frac{2}{3} u^{3/2} + C \).
Here's how to apply the power rule for integration:
For any real number \( n eq -1 \), the rule states: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
In our exercise, after applying substitution, the integral became \( \int \sqrt{u} \, du \), which can be rewritten as \( \int u^{1/2} \, du \). Here, the power rule shines. Setting \( n = \frac{1}{2} \), we integrated to get \( \frac{u^{3/2}}{3/2} + C \), simplifying it further to \( \frac{2}{3} u^{3/2} + C \).
Here's how to apply the power rule for integration:
- Identify your function as a power of \( x \).
- Apply the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
- Simplify the result to arrive at the function’s antiderivative.
- Don’t forget to include \( C \), as it represents the family of antiderivatives.
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