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image_path = '/mnt/data/1733564639374v76r1itb.jpg' image = Image.open(image_path) image.show()I've displayed the image you've uploaded. Let me now analyze its contents to assist with your mathematical question. pytesseract

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extracted_text = pytesseract.image_to_string(image) extracted_textThe image contains the following problem:

Using trigonometric substitution, evaluate the following integral:

$\int \frac{dx}{(81 + x^2)^2}$

This is an integral that can be solved using a trigonometric substitution. To evaluate this, we will use the substitution method where we set $x = 9 \tan(\theta)$ to simplify the denominator.

Steps for solving:

1. Substitute $x = 9 \tan(\theta)$, hence $dx = 9 \sec^2(\theta) \, d\theta$.
2. Substitute into the integral: $\int \frac{9 \sec^2(\theta) \, d\theta}{(81 + (9 \tan(\theta))^2)^2}$
3. Simplify the expression, using the identity $1 + \tan^2(\theta) = \sec^2(\theta)$, to rewrite the integrand in terms of $\sec(\theta)$.
4. Integrate, then back-substitute $\theta$ to express the result in terms of $x$.

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