Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 3 sec^4(3x) dx

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Recognize that the integral involves a power of secant function: \(\int 3 \sec^4(3x) \, dx\). The constant 3 can be factored out, so rewrite the integral as \(3 \int \sec^4(3x) \, dx\).

Recall the reduction formula for powers of secant: for \(n > 1\), \(\int \sec^n(x) \, dx = \frac{\sec^{n-2}(x) \tan(x)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(x) \, dx\). This formula helps reduce the power of secant step-by-step.

Apply the reduction formula with \(n=4\) to the integral \(\int \sec^4(3x) \, dx\). This gives: \(\int \sec^4(3x) \, dx = \frac{\sec^2(3x) \tan(3x)}{3} + \frac{2}{3} \int \sec^2(3x) \, dx\).

Next, evaluate \(\int \sec^2(3x) \, dx\). Recall that \(\int \sec^2(u) \, du = \tan(u) + C\). Use substitution \(u = 3x\), so \(du = 3 dx\), and adjust the integral accordingly.

Combine all parts: multiply the entire expression by 3 (from the original factor), substitute back the evaluated integral of \(\sec^2(3x)\), and include the constant of integration \(+ C\) at the end.

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주요 개념

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Reduction Formulas

Reduction formulas are recursive relationships that express an integral involving a power of a function in terms of an integral with a lower power. They simplify complex integrals by breaking them down step-by-step, often used for powers of trigonometric functions like secant.

Integration of Secant Functions

Integrating powers of secant functions requires special techniques because secant is not straightforward to integrate directly. Common methods include using identities, substitution, or reduction formulas to reduce the power and evaluate the integral.

Substitution Method in Integration

The substitution method involves changing variables to simplify an integral. For integrals like ∫ sec^n(ax) dx, substituting u = ax helps manage constants and makes applying reduction formulas or other techniques easier.

Use reduction formulas to evaluate the integrals in Exercises 41–50.∫ 3 sec^4(3x) dx