Evaluate the integrals in Exercises 33–52.
∫ sec⁴(x) tan²(x) dx

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Recall the trigonometric identities: \(\sec^2(x) = 1 + \tan^2(x)\) and the derivative \(\frac{d}{dx}[\tan(x)] = \sec^2(x)\), which will be useful for substitution.

Rewrite the integral \(\int \sec^4(x) \tan^2(x) \, dx\) as \(\int \sec^2(x) \cdot \sec^2(x) \tan^2(x) \, dx\) to separate one \(\sec^2(x)\) factor for substitution.

Express \(\sec^2(x)\) in terms of \(\tan(x)\) using the identity \(\sec^2(x) = 1 + \tan^2(x)\), so the integral becomes \(\int \sec^2(x) \tan^2(x) (1 + \tan^2(x)) \, dx\).

Use the substitution \(u = \tan(x)\), which implies \(du = \sec^2(x) \, dx\). Replace \(\tan(x)\) and \(\sec^2(x) dx\) in the integral accordingly to rewrite it entirely in terms of \(u\).

After substitution, the integral becomes \(\int u^2 (1 + u^2) \, du\). Expand the integrand to \(\int (u^2 + u^4) \, du\) and then integrate term-by-term.

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주요 개념

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

Trigonometric Identities

Trigonometric identities like sec²(x) = 1 + tan²(x) help simplify integrals involving powers of secant and tangent. Using these identities allows rewriting the integrand into a more manageable form for integration.

Integration by Substitution

Integration by substitution involves changing variables to simplify the integral. For integrals with secant and tangent, substituting u = tan(x) often transforms the integral into a polynomial form, making it easier to solve.

Reduction of Powers in Trigonometric Integrals

Reducing powers means expressing higher powers of trig functions in terms of lower powers or simpler functions. This technique is essential for integrating expressions like sec⁴(x) tan²(x), breaking them down into integrable parts.

Evaluate the integrals in Exercises 33–52.∫ sec⁴(x) tan²(x) dx