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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.1.36
Chapter 8, Problem 8.1.36

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dx / ((2x + 1)√(4x + 4x²)))

Verified step by step guidance
1
First, rewrite the integral to clearly see the expression inside the square root: \(\int \frac{dx}{(2x + 1) \sqrt{4x + 4x^{2}}}\).
Notice that the expression inside the square root can be factored: \(4x + 4x^{2} = 4x(1 + x)\). So the integral becomes \(\int \frac{dx}{(2x + 1) \sqrt{4x(1 + x)}}\).
Simplify the square root: \(\sqrt{4x(1 + x)} = 2 \sqrt{x(1 + x)}\). Substitute this back into the integral to get \(\int \frac{dx}{(2x + 1) \cdot 2 \sqrt{x(1 + x)}} = \int \frac{dx}{2(2x + 1) \sqrt{x(1 + x)}}\).
To simplify the integral, consider a substitution that will simplify the square root term. For example, let \(t = \sqrt{\frac{x}{1 + x}}\) or try a substitution for \(x\) in terms of a new variable to simplify \(\sqrt{x(1 + x)}\). Alternatively, try the substitution \(u = \sqrt{x(1 + x)}\) and express \(dx\) and \(x\) in terms of \(u\).
After choosing an appropriate substitution, rewrite the integral entirely in terms of the new variable, simplify the resulting expression, and then integrate using standard integral techniques such as partial fractions or trigonometric substitution if necessary.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can often be transformed into a more manageable form. This method is especially useful when the integrand contains composite functions.
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Substitution With an Extra Variable

Simplifying Radicals and Algebraic Expressions

Simplifying expressions under the square root and factoring polynomials helps to rewrite the integral in a simpler form. Recognizing perfect squares or factoring quadratics can reduce complexity, making substitution or other integration techniques easier to apply.
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Simplifying Trig Expressions

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions, leveraging identities to simplify the integral. This technique is useful when the integrand contains expressions like √(a² ± x²), enabling easier integration through known trigonometric integrals.
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Introduction to Trigonometric Functions
Related Practice
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