$$ \int_0^{\frac{\pi}{5}}\!\frac{\left(1+\cos^2\theta^{}\right)}{\cos^2\theta}\,\mathrm{d\theta} $$

To evaluate the integral

\[ \int_0^{\frac{\pi}{5}} \frac{1+\cos^2 \theta}{\cos^2 \theta} \, \mathrm{d} \theta, \]

we start by simplifying the integrand:

\[ \frac{1+\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} + 1. \]

Recognizing that $\frac{1}{\cos^2 \theta} = \sec^2 \theta$, we can rewrite the integral as:

\[ \int_0^{\frac{\pi}{5}} \sec^2 \theta \, \mathrm{d} \theta + \int_0^{\frac{\pi}{5}} 1 \, \mathrm{d} \theta. \]

Now, we can evaluate each integral separately.

1. **First Integral:**

\[ \int_0^{\frac{\pi}{5}} \sec^2 \theta \, \mathrm{d} \theta \]

The antiderivative of $\sec^2 \theta$ is $\tan \theta$. Thus,

\[ \int_0^{\frac{\pi}{5}} \sec^2 \theta \, \mathrm{d} \theta = \left[ \tan \theta \right]_0^{\frac{\pi}{5}} = \tan\left(\frac{\pi}{5}\right) - \tan(0) = \tan\left(\frac{\pi}{5}\right). \]

2. **Second Integral:**

\[ \int_0^{\frac{\pi}{5}} 1 \, \mathrm{d} \theta = \left[ \theta \right]_0^{\frac{\pi}{5}} = \frac{\pi}{5} - 0 = \frac{\pi}{5}. \]

Putting it all together, we have:

\[ \int_0^{\frac{\pi}{5}} \frac{1+\cos^2 \theta}{\cos^2 \theta} \, \mathrm{d} \theta = \tan\left(\frac{\pi}{5}\right) + \frac{\pi}{5}. \]

Therefore, the value of the integral is:

\[ \boxed{\tan\left(\frac{\pi}{5}\right) + \frac{\pi}{5}}. \]

Would you like to see more details about this or have any other questions?

Here are 5 related questions you might be interested in:
1. What is the integral of $\sec^2 \theta$ over a different interval?
2. How do you integrate functions involving trigonometric identities?
3. Can you explain the properties of the tangent function in more detail?
4. How do you find the antiderivative of $\sec \theta$?
5. What are some common techniques for simplifying integrals?

**Tip:** When dealing with integrals of trigonometric functions, try to simplify the integrand using trigonometric identities to make the integral easier to evaluate.

Learn how to evaluate the integral of (1 + cos^2(theta)) / cos^2(theta) from 0 to pi/5 with a detailed step-by-step solution. This problem involves integral calculus and trigonometric identities, specifically focusing on the secant function and its antiderivative. Suitable for college-level students and mathematics enthusiasts.

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