6.2
部分積分：定積分
固定区間における2つの関数の積の定積分は、部分積分を用いて評価できます。この方法では、積分は、端点で評価した積と、多くの場合より計算が簡単な残りの定積分との差として書き換えられます。
代表的な例として、逆正接関数の定積分が挙げられます。arctan x に対しては、直接的な初等関数による積分公式が存…
固定区間上の2つの関数の積の定積分は、部分積分を用いて解くことができます。
この式の右辺を解くために、区間の端点間の関数の積の差を評価し、残りの項を定積分として扱います。
有用な例として逆接関数の積分があります。この関数には標準的な積分式が存在しないため、積分関数は逆接と定数1の積として扱われます。
逆接線を微分する関数とし、定数を積分します。
部分積分式に代入すると、最初の項は端点で積を直接評価することで解くことができます。残りの積分は代入によって解くことができます。
新しい変数 tを1× x の二乗に設定し、積分の極限を調整します。積分は対数形式に簡略化される逆式となります。
自然対数は1がゼロであるため、最終的な式は2つの極限の間の曲線下の面積を示すように簡略化されます。
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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.