📚 Integration Methods: The Power Plus One Rule | 积分方法：至少一项幂次加1
The power rule for integration is the first major antidifferentiation technique students meet. It says that when integrating a term like xⁿ, you increase the exponent by 1 and divide by that new exponent. Because the exponent must change from n to n+1, this rule is often called the "power plus one" rule.
积分幂法则是学生接触的第一个重要求原函数技巧。它告诉我们：当对形如 xⁿ 的项积分时，把指数增加 1，再除以这个新指数。由于指数必须从 n 变为 n+1，这一法则常被称为“幂次加一”法则。
1. The Power Rule for Integration | 积分幂法则
For any real number n, as long as n ≠ -1, the integral of xⁿ with respect to x is given by the following formula.
对于任意实数 n，只要 n ≠ -1，xⁿ 对 x 的积分由以下公式给出。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
e, ∫ x⁵ dx = x⁶/6 + C. The exponent 5 increases to 6, and we divide by the new exponent 6.
这里的 C 是积分常数，因为微分会消去常数项，所以积分后必须补回 C。例如，∫ x⁵ dx = x⁶/6 + C。指数 5 增加到 6，然后除以新的指数 6。
only value of n that causes a problem is n = -1, which we will examine in a later section.
该法则适用于正指数、负指数，甚至分数指数。唯一会出现问题的 n 值是 n = -1，我们会在后面的小节中专门讨论。
2. Why Power Plus One? | 为什么幂次加一？
Differentiation normally reduces the exponent by 1. For example, the derivative of xⁿ is n xⁿ⁻¹. Integration is the reverse process, so to undo differentiation we must increase the exponent by 1.
微分通常会把指数减少 1。例如，xⁿ 的导数是 n xⁿ⁻¹。积分是微分的逆过程，所以要“撤销”微分，就必须把指数增加 1。
d/dx [xⁿ⁺¹/(n+1)] = (n+1)xⁿ/(n+1) = xⁿ
This is a quick check: if we differentiate xⁿ⁺¹/(n+1), the factor n+1 cancels, leaving exactly xⁿ. Therefore integration of xⁿ must produce a result whose derivative is xⁿ, and xⁿ⁺¹/(n+1) is the correct antiderivative.
这是一个快速验证：如果对 xⁿ⁺¹/(n+1) 求导，因子 n+1 会约去，正好剩下 xⁿ。因此 xⁿ 的积分结果必须是导数为 xⁿ 的函数，而 xⁿ⁺¹/(n+1) 正是正确的原函数。
| Expression | Derivative | Integral |
| x³ | 3x² | x⁴/4 + C |
| x² | 2x | x³/3 + C |
| x | 1 | x²/2 + C |
The table shows the pattern clearly: differentiation makes the power fall by 1, while integration makes the power rise by 1.
表格清楚地展示了这一规律：微分使幂次下降 1，积分使幂次上升 1。
3. The Exception: n = -1 | 例外：n = -1
re division by n+1 = 0. Since division by zero is undefined, the power rule fails for x⁻¹.
如果尝试对 n = -1 使用幂法则，公式中会出现除以 n+1 = 0 的情况。由于除以零没有意义，幂法则对 x⁻¹ 失效。
∫ x⁻¹ dx = ln|x| + C
The integral of 1/x is the natural logarithm of the absolute value of x, plus C. The absolute value is essential because ln x is only defined for positive x, while 1/x is defined for both positive and negative x.
1/x 的积分是自然对数 ln|x|，再加 C。绝对值符号非常重要，因为 ln x 只对正数 x 有定义，而 1/x 对正数和负数 x 都有定义。
This single exception is worth memorising. Whenever you see 1/x or x⁻¹ in an integral, do not apply the power rule; use ln|x| + C instead.
这个唯一的例外值得牢记。只要在积分中看到 1/x 或 x⁻¹，就不要使用幂法则，而应使用 ln|x| + C。
4. Integrating Polynomials Term by Term | 多项式逐项积分
A polynomial is a sum of power terms. Because integration is linear, we can integrate each term separately and then add the results.
多项式是若干幂项之和。由于积分具有线性性质，我们可以逐项积分，再把结果相加。
∫ (3x² + 2x + 1) dx = x³ + x² + x + C
Let us check each term. For 3x², the power 2 becomes 3, and we divide by 3, giving x³. For 2x, the power 1 becomes 2, and we divide by 2, giving x². For the constant 1, think of it as 1x⁰; the power 0 becomes 1, giving x.
让我们逐项验证。对于 3x²，指数 2 变为 3，除以 3，得到 x³。对于 2x，指数 1 变为 2，除以 2，得到 x²。对于常数 1，可以把它看成 1x⁰；指数 0 变为 1，得到 x。
Notice that every term's exponent has increased by exactly 1. No term is left with its original exponent. This is why the phrase "at least one power plus one" is commonly used when checking polynomial integrations.
注意，每一项的指数都恰好增加了 1。没有任何一项保留原来的指数。这就是为什么“至少一项幂次加1”常被用来检查多项式积分是否正确的说法。
5. Constants and Coefficients | 常数与系数的处理
When a term has a constant coefficient, that coefficient simply stays in front of the result. The power rule only affects the variable part.
当某一项带有常数系数时，这个系数只需保留在结果前面。幂法则只影响变量部分。
∫ 7x³ dx = 7∫ x³ dx = 7 x⁴/4 + C = 7x⁴/4 + C
for any constant a, ∫ a f(x) dx = a ∫ f(x) dx. This is called the constant multiple rule.
一般地，对于任意常数 a，∫ a f(x) dx = a ∫ f(x) dx。这就是常数倍法则。
Be careful not to confuse this with differentiation. When differentiating, a coefficient multiplies the new power; when integrating, the coefficient also multiplies the new term, but the exponent still increases by 1.
注意不要与微分混淆。微分时，系数会乘到新的幂项前；积分时，系数同样乘到新项前，但指数仍然增加 1。
6. Negative Powers |
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