6.2
分部积分法：定积分
在固定区间上，对两个函数乘积形式的定积分可采用分部积分法进行计算。该方法将原定积分转化为在区间端点处计算的函数乘积与一个剩余定积分之差，而后者通常结构更为简单，因而更易求解。
一个具有代表性的例子是反正切函数的定积分。由于 arctan x 缺乏初等原函数，可将被积函数表示为 arctan x 与常…
在固定区间上，两个函数乘积的定积分可以通过分部积分法求解。
为求解该表达式的右侧，将在区间端点处计算函数乘积的差值，其余项则作为定积分进行处理。
一个有用的示例是反正切函数的积分。由于该函数没有标准的积分公式，因此将被积函数视为反正切函数与常数1的乘积来处理。
将反切线函数作为待求导函数，而常数则进行积分。
代入分部积分公式后，第一项可通过直接计算端点处的乘积求解。剩余的积分则可通过变量代换法求解。
设一个新变量 t 等于 1 加上 x 的平方，并调整积分限。此时，该积分转化为一个倒数表达式，可简化为对数形式。
1 的自然对数等于零，因此最终表达式可简化为表示两条极限之间曲线下方的面积。
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Q1: How does integration by parts apply to definite integrals?
Integration by parts for definite integrals rewrites the integral as the difference of a product evaluated at the endpoints minus a remaining definite integral. The formula transforms products of two functions into simpler forms by choosing which function to differentiate and which to integrate, then evaluating the product term at the interval's boundaries.
Q2: Why is the inverse tangent function difficult to integrate directly?
The inverse tangent function has no standard integration formula, so it cannot be integrated using elementary antiderivative techniques. To solve integrals involving arctan(x), the integrand is rewritten as a product of the inverse tangent and the constant 1, allowing integration by parts to be applied effectively.
Q3: What role does substitution play after applying integration by parts?
After integration by parts simplifies the original integral, substitution is used to evaluate the remaining definite integral. By introducing a new variable such as t = 1 + x², the integral is transformed into a reciprocal form that integrates to a logarithmic expression, with limits adjusted accordingly for the new variable.
Q4: How do you evaluate the product term in the integration by parts formula?
The product term is evaluated by computing the product of the two chosen functions at each endpoint of the integration interval, then finding the difference between these values. This direct evaluation eliminates the need to find an antiderivative for the product itself.
Q5: Why does the natural logarithm of one equal zero in the final answer?
The natural logarithm of one equals zero by definition, since e^0 = 1. When evaluating logarithmic expressions at the bounds of a definite integral, this property simplifies the result, often eliminating terms and leaving a cleaner final expression for the area under the curve.
Q6: What does the final result of integrating inverse tangent represent?
The final result represents the area under the inverse tangent curve between the given integration limits. This geometric interpretation shows how integration by parts successfully evaluates definite integrals of functions without elementary antiderivatives, providing both a numerical answer and conceptual understanding.
Q7: How does integration by parts for definite integrals differ from integration by parts for indefinite integrals?
Definite integrals include fixed endpoints that are substituted directly into the product term, eliminating the constant of integration. With indefinite integrals, the constant of integration remains in the final answer. Both methods use the same formula structure, but definite integrals yield numerical results while indefinite integrals produce families of antiderivatives.