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

검증된 단계별 안내

Start with the integral \( \int ((x^{2} - 1)(x + 1))^{-\frac{2}{3}} \, dx \). First, simplify the expression inside the integral if possible. Notice that \( (x^{2} - 1)(x + 1) = (x - 1)(x + 1)(x + 1) = (x - 1)(x + 1)^{2} \). So the integral becomes \( \int \left( (x - 1)(x + 1)^{2} \right)^{-\frac{2}{3}} \, dx \).

Apply the substitution \( u = \arctan x \). Then, differentiate both sides to find \( du \) in terms of \( dx \): \( du = \frac{1}{1 + x^{2}} \, dx \), which implies \( dx = (1 + x^{2}) \, du \).

Express the integral entirely in terms of \( u \). Since \( x = \tan u \), rewrite all parts of the integrand: \( x - 1 = \tan u - 1 \), \( x + 1 = \tan u + 1 \), and \( dx = (1 + \tan^{2} u) \, du = \sec^{2} u \, du \). Substitute these into the integral.

Rewrite the integral as \( \int \left( (\tan u - 1)(\tan u + 1)^{2} \right)^{-\frac{2}{3}} \sec^{2} u \, du \). Simplify the expression inside the parentheses if possible, and then simplify the powers and factors to get the integral in terms of \( u \) only.

Once the integral is expressed in terms of \( u \), proceed to integrate with respect to \( u \). After finding the antiderivative, substitute back \( u = \arctan x \) to express the answer in terms of \( x \).

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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 new variable u = g(x) to rewrite the integral in terms of u, making it easier to evaluate. The derivative du/dx replaces dx, and the integral limits or integrand are adjusted accordingly.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions with trigonometric functions to simplify integrals involving roots or powers. For example, using u = arctan(x) transforms expressions involving x into functions of u, leveraging identities like 1 + tan²u = sec²u to simplify the integrand.

Evaluating Integrals with Inverse Trigonometric Functions

When the substitution involves inverse trigonometric functions, the integral is expressed in terms of u = arctan(x). This requires rewriting x and dx in terms of u and using trigonometric identities to simplify the integrand before integrating, often resulting in expressions involving trigonometric or algebraic functions of u.

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