Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx

검증된 단계별 안내

Start by rewriting the integral to clearly see the expression: \(\int \frac{1}{x^{6}(x^{5} + 4)} \, dx\).

Consider a substitution to simplify the integral. Notice that \(x^{5}\) appears inside the parentheses, so let \(u = x^{5} + 4\).

Differentiate \(u\) with respect to \(x\) to find \(du\): \(du = 5x^{4} \, dx\), which implies \(dx = \frac{du}{5x^{4}}\).

Rewrite the integral in terms of \(u\) and \(x\): substitute \(x^{5} + 4\) with \(u\) and \(dx\) with \(\frac{du}{5x^{4}}\). Also, express the remaining powers of \(x\) in terms of \(u\) if possible.

Simplify the integral after substitution and look for further algebraic manipulation or partial fraction decomposition to integrate with respect to \(u\).

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주요 개념

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

Integration of Rational Functions

This involves integrating functions expressed as the ratio of two polynomials. Techniques often include algebraic manipulation, partial fraction decomposition, or substitution to simplify the integral into manageable parts.

Partial Fraction Decomposition

A method used to break down complex rational expressions into simpler fractions that are easier to integrate. It requires factoring the denominator and expressing the integrand as a sum of simpler rational terms.

Substitution Method

A technique where a part of the integrand is replaced with a new variable to simplify the integral. It is especially useful when the integral contains a composite function or when the derivative of a function appears within the integrand.

Evaluate the integrals in Exercises 39–54.∫ 1 / (x⁶(x⁵ + 4)) dx