Standard Integrals Integrating to Inverse Trigonometric Functions | 积分得到反三角函数的常用积分

📚 Standard Integrals Integrating to Inverse Trigonometric Functions | 积分得到反三角函数的常用积分

When we differentiate inverse trigonometric functions, we obtain algebraic expressions involving square roots and quadratic denominators. Reversing these differentiations gives us standard integrals that are vital in AQA A-Level Mathematics. In this article, we explore the two principal standard integrals that integrate to arcsin and arctan, along with worked examples and common pitfalls.

当我们对反三角函数求导时，会得到含有根号与二次分母的代数式。将这个过程反过来，就得到了 AQA A-Level 数学中非常重要的标准积分。本文将探讨两个主要的、积分结果为 arcsin 和 arctan 的标准积分，并结合例题与常见错误进行分析。

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1. Why Inverse Trigonometric Functions Appear in Integration | 为什么积分中会出现反三角函数

Integration is the reverse process of differentiation. Since the derivatives of arcsin(x) and arctan(x) are algebraic, the corresponding antiderivatives are inverse trigonometric functions. These integrals are not just theoretical curiosities; they appear in problems involving areas under curves, kinematics, and geometry.

积分是微分的逆运算。由于 arcsin(x) 和 arctan(x) 的导数都是代数式，所以对应的原函数就是反三角函数。这些积分不仅是理论上的有趣结论，还出现在曲线下面积、运动学以及几何问题中。

The two fundamental derivatives we need are:

我们需要的两个基本导数是：

d/dx [arcsin(x)] = 1/√(1 − x²)

d/dx [arctan(x)] = 1/(1 + x²)

By applying the chain rule, we obtain more general forms that lead to the standard integrals we use in AQA.

通过链式法则，我们可以得到更一般的形式，从而导出 AQA 考试中使用的标准积分。

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2. Derivatives of arcsin(x/a) and arctan(x/a) | arcsin(x/a) 与 arctan(x/a) 的导数

Consider a constant a > 0. Differentiating arcsin(x/a) with respect to x gives:

考虑常数 a > 0。对 arcsin(x/a) 关于 x 求导得到：

d/dx [arcsin(x/a)] = 1/√(a² − x²)

Similarly, differentiating arctan(x/a) gives:

类似地，对 arctan(x/a) 求导得到：

d/dx [arctan(x/a)] = a/(a² + x²)

These two derivatives are the direct source of the standard integrals. Reversing them, we obtain the integrals listed in the AQA formula booklet.

这两个导数正是标准积分的直接来源。将它们反转，便得到了 AQA 公式手册中所列出的积分。

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3. Standard Integral 1: ∫ 1/√(a² − x²) dx | 标准积分 1：∫ 1/√(a² − x²) dx

The first key standard integral is:

第一个关键标准积分是：

∫ 1/√(a² − x²) dx = arcsin(x/a) + C

This is valid for a > 0 and |x| < a, where the expression under the square root is positive.

该公式适用于 a > 0 且 |x| < a，此时根号下的表达式为正数。

Notice that the result is independent of a’s value in the numerator; the constant a appears only inside arcsine. This is because the derivative of arcsin(x/a) automatically produces the necessary factor.

注意结果中 a 并不出现在分母；常数 a 只出现在 arcsin 内部。这是因为 arcsin(x/a) 的导数恰好产生了所需的因子。

If a = 1, we recover the simpler form:

如果 a = 1，我们就得到简化形式：

∫ 1/√(1 − x²) dx = arcsin x + C

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4. Worked Example 1 | 例题 1

Evaluate ∫ 1/√(9 − x²) dx.

计算 ∫ 1/√(9 − x²) dx。

Here, a² = 9, so a = 3. Using the standard integral:

这里 a² = 9，因此 a = 3。利用标准积分：

∫ 1/√(9 − x²) dx = arcsin(x/3) + C

It is that simple. You only need to recognise the exact form 1/√(a² − x²).

就是这么简单。你只需要识别出 1/√(a² − x²) 的确切形式。

For a definite integral, for example ∫₀³ 1/√(9 − x²) dx, we evaluate:

对于定积分，例如 ∫₀³ 1/√(9 − x²) dx，我们计算：

[arcsin(x/3)]₀³ = arcsin(1) − arcsin(0) = π/2 − 0 = π/2

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5. Standard Integral 2: ∫ 1/(a² + x²) dx | 标准积分 2：∫ 1/(a² + x²) dx

The second key standard integral is:

第二个关键标准积分是：

∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C

Note the factor 1/a in front of arctan. This arises because the derivative of arctan(x/a) gives a/(a² + x²), and to obtain 1/(a² + x²) we must divide by a.

注意 arctan 前面的因子 1/a。这是因为 arctan(x/a) 的导数是 a/(a² + x²)，为了得到 1/(a² + x²)，我们必须除以 a。

When a = 1, this becomes:

当 a = 1 时，变为：

∫ 1/(1 + x²) dx = arctan x + C

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6. Worked Example 2 | 例题 2

Evaluate ∫ 1/(4 + x²) dx.

计算 ∫ 1/(4 + x²) dx。

Here, a² = 4, so a = 2. Hence:

这里 a² = 4，所以 a = 2。于是：

∫ 1/(4 + x²) dx = (1/2) arctan(x/2) + C

Compare this with ∫ 1/(1 + x²) dx = arctan x + C. The only difference is the factor (1/2).

将其与 ∫ 1/(1 + x²) dx = arctan x + C 比较，唯一区别就是因子 (1/2)。

If we instead have ∫ 3/(9 + x²) dx, we can factor out the constant 3:

如果题目是 ∫ 3/(9 + x²) dx，我们可以提取常数 3：

3 ∫ 1/(9 + x²) dx = 3 × (1/3) arctan(x/3) + C = arctan(x/3) + C

Notice how the factor 3 cancels neatly. Always check coefficients before integrating.

注意到常数 3 被巧妙地约去了。积分前务必检查系数。

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7. Combining the Forms | 组合使用两种形式

Some integrands contain both patterns or require simple algebraic manipulation. For example, consider ∫ (x + 1)/(x² + 1) dx.

有些被积函数同时包含两种形式，或者需要简单的代数变形。例如，考虑 ∫ (x + 1)/(x² + 1) dx。

Split the fraction:

将分式拆开：

∫ x/(x² + 1) dx + ∫ 1/(x² + 1) dx

The second part is arctan x. The first part is handled by substitution u = x² + 1, giving (1/2) ln(x² + 1). So:

第二部分是 arctan x。第一部分用换元 u = x² + 1 处理，得到 (1/2) ln(x² + 1)。所以：

∫ (x + 1)/(x² + 1) dx = (1/2) ln(x² + 1) + arctan x + C

This example shows that inverse trig integrals often appear alongside logarithms.

这个例子表明，反三角函数积分经常和对数一同出现。

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8. Completing the Square | 配方法的应用

When the denominator is a quadratic that does not match the simple x² ± a² pattern, we can complete the square. For instance, to integrate ∫ 1/(x² + 2x + 5) dx, rewrite the denominator:

当分母是一个不符合简单 x² ± a² 形式的二次式时，我们可以使用配方法。例如，要计算 ∫ 1/(x² + 2x + 5) dx，将分母改写：

x² + 2x + 5 = (x + 1)² + 4

Let u = x + 1. Then du = dx, and the integral becomes:

令 u = x + 1，则 du = dx，积分变为：

∫ 1/(u² + 4) du = (1/2) arctan(u/2) + C = (1/2) arctan((x + 1)/2) + C

Completing the square is an essential skill for turning unfamiliar quadratics into one of the two standard forms.

配方法是将陌生二次式转化为两种标准形式的关键技能。

Similarly, for ∫ 1/√(2x − x²) dx, rewrite 2x − x² = 1 − (x − 1)², which matches the arcsin form with a = 1 and shift u = x − 1.

类似地，对于 ∫ 1/√(2x − x²) dx，将 2x − x² 改写为 1 − (x − 1)²，这符合 a = 1 的 arcsin 形式，并平移 u = x − 1。

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9. Definite Integrals | 定积分

When evaluating a definite integral, remember to substitute the limits into the antiderivative and subtract carefully. For example:

计算定积分时，记得将上下限代入原函数并仔细相减。例如：

Evaluate ∫₀¹ 1/(1 + x²) dx.

计算 ∫₀¹ 1/(1 + x²) dx。

[arctan x]₀¹ = arctan 1 − arctan 0 = π/4 − 0 = π/4

In AQA problems, the limits often involve special values of arctan or arcsin, giving exact answers with π.

在 AQA 题目中，上下限常涉及 arctan 或 arcsin 的特殊值，从而得到含有 π 的精确答案。

Be careful with negative limits: arctan is an odd function, so arctan(−a) = −arctan(a).

注意负数限：arctan 是奇函数，因此 arctan(−a) = −arctan(a)。

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10. Pitfalls and Exam Tips | 常见错误与考试技巧

Students often confuse the two standard integrals. The key difference is the sign inside the denominator: minus for arcsin, plus for arctan.

学生经常混淆这两个标准积分。关键区别在于分母中的符号：√(a² − x²) 对应 arcsin，a² + x² 对应 arctan。

A common mistake is forgetting the factor 1/a in the arctan integral. Always check: ∫ 1/(a² + x²) dx has 1/a in front of the result.

一个常见错误是忘记 arctan 积分中的 1/a 因子。务必检查：∫ 1/(a² + x²) dx 的结果前有 1/a。

When the integrand looks like 1/√(k² − x²), identify k as the positive square root of the constant term. For example, 1/√(25 − x²) integrates to arcsin(x/5), not arcsin(5x).

当被积函数形如 1/√(k² − x²) 时，k 是常数项的正平方根。例如，1/√(25 − x²) 的积分是 arcsin(x/5)，而不是 arcsin(5x)。

Always add the constant of integration C for indefinite integrals.

对于不定积分，不要忘记添加积分常数 C。

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11. Summary Table | 标准积分汇总表

Integral	Result
∫ 1/√(a² − x²) dx	arcsin(x/a) + C
∫ 1/(a² + x²) dx	(1/a) arctan(x/a) + C
∫ 1/√(1 − x²) dx	arcsin x + C
∫ 1/(1 + x²) dx	arctan x + C

These four forms are the ones AQA students should memorise. The general forms with a are used when a does not equal 1.

以上四种形式是 AQA 学生应当牢记的。当 a ≠ 1 时，就需要使用带 a 的一般形式。

Once you master these standard integrals and the technique of completing the square, you will be able to handle a wide range of AQA integration questions confidently.

一旦掌握了这些标准积分以及配方法，你就能从容应对 AQA 中大量的积分题目。

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