Math Problem Statement

0π5 ⁣(1+cos2θ)cos2θdθ\int_0^{\frac{\pi}{5}}\!\frac{\left(1+\cos^2\theta^{}\right)}{\cos^2\theta}\,\mathrm{d\theta}

Solution

To evaluate the integral

0π51+cos2θcos2θdθ,\int_0^{\frac{\pi}{5}} \frac{1+\cos^2 \theta}{\cos^2 \theta} \, \mathrm{d} \theta,

we start by simplifying the integrand:

1+cos2θcos2θ=1cos2θ+1.\frac{1+\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} + 1.

Recognizing that 1cos2θ=sec2θ\frac{1}{\cos^2 \theta} = \sec^2 \theta, we can rewrite the integral as:

0π5sec2θdθ+0π51dθ.\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:

0π5sec2θdθ\int_0^{\frac{\pi}{5}} \sec^2 \theta \, \mathrm{d} \theta

The antiderivative of sec2θ\sec^2 \theta is tanθ\tan \theta. Thus,

0π5sec2θdθ=[tanθ]0π5=tan(π5)tan(0)=tan(π5).\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).

  1. Second Integral:

0π51dθ=[θ]0π5=π50=π5.\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:

0π51+cos2θcos2θdθ=tan(π5)+π5.\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:

tan(π5)+π5.\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 sec2θ\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 secθ\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.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Trigonometric Identities

Formulas

Integral of sec^2(theta)
Antiderivative of sec^2(theta)

Theorems

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Suitable Grade Level

College Level