Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(4x) / (x³ + 4x)] dx

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Start by simplifying the integrand \( \frac{4x}{x^3 + 4x} \). Notice that the denominator can be factored as \( x(x^2 + 4) \), so rewrite the integral as \( \int \frac{4x}{x(x^2 + 4)} \, dx \).

Cancel the common factor \( x \) in the numerator and denominator, assuming \( x \neq 0 \), to simplify the integrand to \( \int \frac{4}{x^2 + 4} \, dx \).

Recognize that the integral now has the form \( \int \frac{C}{x^2 + a^2} \, dx \), which is a standard integral. Recall the formula: \( \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left( \frac{x}{a} \right) + C \).

Apply the constant multiple rule to factor out the 4, so the integral becomes \( 4 \int \frac{1}{x^2 + 4} \, dx \). Here, \( a^2 = 4 \), so \( a = 2 \).

Use the formula to write the integral as \( 4 \times \frac{1}{2} \arctan\left( \frac{x}{2} \right) + C \), simplifying the constants as needed.

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주요 개념

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.

Simplifying Rational Expressions

Simplifying rational expressions involves factoring and reducing fractions to make integration easier. In this problem, factoring the denominator can reveal common factors with the numerator, allowing simplification before integrating. This step often clarifies the integral's structure.

Basic Integration Rules

Basic integration rules include integrating powers of x, constants, and sums or differences of functions. After substitution and simplification, applying these fundamental rules allows the evaluation of the integral. Familiarity with these rules is essential for solving integrals efficiently.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [(4x) / (x³ + 4x)] dx