Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-sec²(3x/2)

검증된 단계별 안내

Recognize that the function given is \(-\sec^{2}\left(\frac{3x}{2}\right)\), which resembles the derivative of the tangent function, since \(\frac{d}{dx}[\tan(u)] = \sec^{2}(u) \cdot \frac{du}{dx}\).

Identify the inner function \(u = \frac{3x}{2}\) and compute its derivative: \(\frac{du}{dx} = \frac{3}{2}\).

Set up the antiderivative integral: \(\int -\sec^{2}\left(\frac{3x}{2}\right) dx\).

Use substitution: let \(u = \frac{3x}{2}\), so \(dx = \frac{2}{3} du\). Rewrite the integral in terms of \(u\): \(\int -\sec^{2}(u) \cdot \frac{2}{3} du = -\frac{2}{3} \int \sec^{2}(u) du\).

Recall that \(\int \sec^{2}(u) du = \tan(u) + C\), so the antiderivative is \(-\frac{2}{3} \tan\left(\frac{3x}{2}\right) + C\).

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주요 개념

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

Antiderivative (Indefinite Integral)

An antiderivative of a function is another function whose derivative equals the original function. It is also called the indefinite integral and includes a constant of integration since differentiation loses constant terms.

Derivative of Trigonometric Functions

Knowing the derivatives of basic trig functions like tan(x) and sec(x) is essential. For example, the derivative of tan(x) is sec²(x), which helps in recognizing antiderivatives involving sec² terms.

Chain Rule and Substitution

When functions involve compositions like sec²(3x/2), the chain rule applies. To find antiderivatives, substitution reverses the chain rule by adjusting for the inner function's derivative.

Finding AntiderivativesIn Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.-sec²(3x/2)