Math Problem Statement

  1. Trig Substitution. Use a trig substitution to evaluate the indefinite integral. Show all the details of your substitution. fdx/x^2 sqrtx^2+1 ANTIDERIVATIVE: answers only

Solution

The integral given is:

∫dxx2x2+1\int \frac{dx}{x^2 \sqrt{x^2 + 1}}

1. Substitution:

Use the substitution x=tan⁡(θ)x = \tan(\theta), which gives: dx=sec⁡2(θ)dθ,x2+1=sec⁡2(θ)dx = \sec^2(\theta) d\theta, \quad x^2 + 1 = \sec^2(\theta)

2. Transform the integral:

Substitute x=tan⁡(θ)x = \tan(\theta) into the integral:

= \int \frac{d\theta}{\tan^2(\theta)} = \int \cot^2(\theta) d\theta$$ ### 3. Simplify and integrate: Use the identity $$\cot^2(\theta) = \csc^2(\theta) - 1$$: $$\int \cot^2(\theta) d\theta = \int (\csc^2(\theta) - 1) d\theta$$ $$= -\cot(\theta) - \theta + C$$ ### 4. Back-substitute $$\theta = \tan^{-1}(x)$$: Since $$\cot(\theta) = \frac{1}{x}$$ and $$\theta = \tan^{-1}(x)$$, we have: $$-\frac{1}{x} - \tan^{-1}(x) + C$$ ### Final Answer: $$\boxed{-\frac{1}{x} - \tan^{-1}(x) + C}$$ Would you like to review any specific details or steps? Here are 5 related questions to explore further: 1. How does the identity $$\cot^2(\theta) = \csc^2(\theta) - 1$$ arise? 2. Can you use different substitutions for the same integral? What happens? 3. How does this method apply to similar integrals involving $$\sqrt{x^2 - 1}$$? 4. What are the conditions for using trigonometric substitution? 5. Can you derive the formulas for the inverse trigonometric functions used here? **Tip:** Always double-check trigonometric identities when simplifying integrals involving substitutions.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Substitution
Integration
Trigonometric Identities

Formulas

x = tan(θ)
dx = sec^2(θ) dθ
cot^2(θ) = csc^2(θ) - 1

Theorems

Trigonometric Substitution
Pythagorean Identities

Suitable Grade Level

Undergraduate Calculus

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