Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
e. u = tan^(-1) ((x - 1)/2)
What is the value of the integral?

검증된 단계별 안내

Start with the integral \( \int ((x^2 - 1)(x + 1))^{-2/3} \, dx \). First, simplify the expression inside the integral. Notice that \( x^2 - 1 = (x - 1)(x + 1) \), so the integrand becomes \( ((x - 1)(x + 1)^2)^{-2/3} \).

Rewrite the integrand as \( (x - 1)^{-2/3} (x + 1)^{-4/3} \) by distributing the exponent \( -2/3 \) to each factor inside the parentheses.

Use the substitution \( u = \tan^{-1} \left( \frac{x - 1}{2} \right) \). This means \( \frac{x - 1}{2} = \tan u \), so \( x - 1 = 2 \tan u \).

Differentiate \( x - 1 = 2 \tan u \) with respect to \( u \) to find \( dx \) in terms of \( du \): \( dx = 2 \sec^2 u \, du \).

Express \( (x + 1) \) in terms of \( u \) using the substitution and simplify the integrand accordingly. Then rewrite the entire integral in terms of \( u \) and \( du \), which will allow you to integrate with respect to \( u \).

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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 to simplify integrals by changing variables. It involves choosing a substitution u = g(x) that transforms the integral into a simpler form in terms of u, making it easier to evaluate. This technique is especially useful when the integrand contains composite functions.

Inverse Trigonometric Substitution

Inverse trigonometric substitution involves substituting a variable with an inverse trigonometric function, such as u = arctan((x - a)/b), to simplify integrals involving algebraic expressions. This substitution leverages trigonometric identities to transform complicated expressions into integrable forms.

Handling Rational Exponents in Integrals

Integrals with rational exponents, like powers of -2/3, require careful manipulation of the integrand. Understanding how to rewrite and simplify expressions with fractional powers is essential, often involving factoring or substitution to convert the integral into a standard form.

Using different substitutionsShow that the integral∫((x² - 1)(x + 1))^(-2/3) dxcan be evaluated with any of the following substitutions.e. u = tan^(-1) ((x - 1)/2)What is the value of the integral?