Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / (x (1 + ∛x))] dx

검증된 단계별 안내

Start by examining the integral \( \int \frac{1}{x (1 + \sqrt[3]{x})} \, dx \). Notice the presence of \( \sqrt[3]{x} \), which suggests a substitution involving \( x^{1/3} \).

Let \( t = \sqrt[3]{x} = x^{1/3} \). Then, express \( x \) in terms of \( t \): \( x = t^3 \).

Differentiate \( x = t^3 \) with respect to \( t \) to find \( dx \): \( dx = 3t^2 \, dt \).

Rewrite the integral in terms of \( t \) by substituting \( x = t^3 \), \( dx = 3t^2 \, dt \), and \( \sqrt[3]{x} = t \). The integral becomes \( \int \frac{1}{t^3 (1 + t)} \cdot 3t^2 \, dt \).

Simplify the integrand: \( \frac{3t^2}{t^3 (1 + t)} = \frac{3}{t (1 + t)} \). Now, the integral is \( \int \frac{3}{t (1 + t)} \, dt \). To integrate, use partial fraction decomposition on \( \frac{1}{t (1 + t)} \) and then integrate term-by-term.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 composite functions or roots.

Properties of Exponents and Roots

Understanding how to manipulate exponents and roots is essential for rewriting expressions like ∛x (the cube root of x) as x^(1/3). This allows for easier algebraic manipulation and substitution during integration, enabling the integral to be expressed in terms of powers of x.

Integration of Rational Functions

Integrating rational functions involves integrating ratios of polynomials or expressions that can be transformed into such ratios. Techniques include partial fraction decomposition or substitution to simplify the integrand, making it easier to find the antiderivative.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [1 / (x (1 + ∛x))] dx