$$\int_{0}^{1} \frac{2x}{1+x^2} dx = ?$$ A) $1$ B) $\ln 2$ C) $2$ D) $\ln 4$

1. How do I solve this?
2. <details class="reasoning-details"> <summary style="cursor:pointer; color:#666; font-size:0.9em;">Reasoning (click to expand)</summary> <div class="reasoning-block"> **Analyzing the Integral** I'm focusing on the relationship between the numerator and the denominator within the integral. My aim is to guide the user toward recognizing a key calculus principle that simplifies this specific expression. </div> </details> Look closely at the denominator, $1 + x^2$, and the numerator, $2x$. What do you get if you take the derivative of the denominator with respect to $x$? How does that relate to what is in the numerator?
3. 这道题怎么做？
4. <details class="reasoning-details"> <summary style="cursor:pointer; color:#666; font-size:0.9em;">思考过程 (点击展开)</summary> <div class="reasoning-block"> **Solving the integral** I'm currently working through a definite integral problem, specifically $\int_{0}^{1} \frac{2x}{1+x^2} dx$. My current approach is to utilize a substitution method, as this seems like the most direct path to a solution given the form of the integrand. </div> </details> 观察被积函数的分母 $1 + x^2$ 和分子 $2x$。 先试着想一想：如果对分母求导，即 $(1 + x^2)'$，结果是多少？它和分子有什么关系？
5. この問題はどう解けばいいですか？
6. <details class="reasoning-details"> <summary style="cursor:pointer; color:#666; font-size:0.9em;">思考プロセス（クリックで展開）</summary> <div class="reasoning-block"> **Exploring the Integral** I'm currently examining the definite integral $\int_{0}^{1} \frac{2x}{1+x^2} dx$. My initial focus is on identifying the most efficient method to solve this, considering the provided options. </div> </details> まずは、被積分関数の**分母** $1 + x^2$ に注目してみましょう。 分母を $x$ で微分する（$(1 + x^2)'$ を計算する）と、何になりますか？ **分子**の $2x$ と見比べて、気づくことはありますか？
7. 私の解き方は合っていますか？
8. <details class="reasoning-details"> <summary style="cursor:pointer; color:#666; font-size:0.9em;">思考プロセス（クリックで展開）</summary> <div class="reasoning-block"> **Analyzing the Calculus Steps** I'm currently reviewing the integration process. I've confirmed the derivative calculation and the substitution for 't'. The limits of integration have also been correctly adjusted based on the substitution. The next step is to complete the integral with these new bounds. </div> </details> **完璧です！正解です。** - $t = 1 + x^2$ と置いたときの微分の対応（$dt = 2x\,dx$） - 積分区間の変換（$x: 0 \to 1$ が $t: 1 \to 2$ に変わる点） - $\log 1 = 0$ の計算 すべて正確にできています。自信を持って大丈夫ですよ！