o1的数学能力相比之前有质的提升1，差不多是从正确率20%多进步到86%的水平，而且这个前提是奥赛题目。那么做做高中水平的积分岂不是砍瓜切菜？
如果有其它想尝试的题目可以在下面留言。
我的提问
Evaluate the integral using trigonometric substitution
\int \frac{1}{y^2 \sqrt{10 - y^2}} \, dy
o1的回答
To evaluate the integral
\int \frac{1}{y^2 \sqrt{10 - y^2}} \, dy,
we can use trigonometric substitution. Here's a step-by-step solution:
Step 1: Choose an Appropriate Substitution
The presence of \sqrt{10 - y^2} suggests using the substitution:
y = \sqrt{10} \sin \theta
This substitution leverages the Pythagorean identity, which simplifies the square root.
Step 2: Compute dy and Simplify the Integral
Differentiate y with respect to \theta:
dy = \sqrt{10} \cos \theta \, d\theta
Substitute y and dy into the integral:
\int \frac{1}{y^2 \sqrt{10 - y^2}} \, dy = \int \frac{1}{10 \sin^2 \theta \cdot \sqrt{10 - 10 \sin^2 \theta}} \cdot \sqrt{10} \cos \theta \, d\theta
Simplify the square root:
\sqrt{10 - 10 \sin^2 \theta} = \sqrt{10 (1 - \sin^2 \theta)} = \sqrt{10} \cos \theta
Substitute back:
\int \frac{1}{10 \sin^2 \theta \cdot \sqrt{10} \cos \theta} \cdot \sqrt{10} \cos \theta \, d\theta = \frac{1}{10} \int \csc^2 \theta \, d\theta
Step 3: Integrate with Respect to \theta
The integral of \csc^2 \theta is straightforward:
\frac{1}{10} \int \csc^2 \theta \, d\theta = -\frac{1}{10} \cot \theta + C
Step 4: Re-substitute \theta in Terms of y
From the original substitution y = \sqrt{10} \sin \theta, we can find \cot \theta:
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\sqrt{10 - y^2}/\sqrt{10}}{y/\sqrt{10}} = \frac{\sqrt{10 - y^2}}{y}
Thus, the integral in terms of y is:
-\frac{1}{10} \cdot \frac{\sqrt{10 - y^2}}{y} + C = -\frac{\sqrt{10 - y^2}}{10 y} + C
Final Answer
After simplifying, an antiderivative is -\sqrt{10 - y^2} divided by 10y plus C. Thus,
-\frac{\sqrt{10 - y^2}}{10 y} + C