3.6 Q-15

Question Statement

Evaluate the following integral: $\int_{0}^{\frac{\pi}{4}} \frac{\cos \theta + \sin \theta}{\cos 2\theta} , d\theta$

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Background and Explanation

This integral involves a combination of trigonometric functions, including $\cos \theta$, $\sin \theta$, and $\cos 2\theta$. A key concept for solving this integral is simplifying the trigonometric terms and recognizing common integrals like those of $\sec \theta$ and $\sec \theta \tan \theta$, which are standard integrals.

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Solution

1. Simplify the Expression:

   Start by simplifying the integrand. We rewrite $\cos 2\theta$ in terms of $\cos \theta$ and $\sin \theta$. From the double angle identity, we have: $\cos 2\theta = 2\cos^2 \theta - 1$

   So, the original integral becomes: $\int_{0}^{\frac{\pi}{4}} \frac{\cos \theta + \sin \theta}{2\cos^2 \theta - 1} , d\theta$

   We now split the numerator into two parts: $\frac{\cos \theta}{\cos 2\theta} + \frac{\sin \theta}{\cos 2\theta}$

2. Separate the Integral:

   Break the integral into two simpler integrals: $\frac{1}{2} \int_{0}^{\frac{\pi}{4}} \frac{\cos \theta}{\cos^2 \theta} , d\theta + \frac{1}{2} \int_{0}^{\frac{\pi}{4}} \frac{\sin \theta}{\cos \theta} , d\theta$

3. Simplify the Terms:

   • The first term simplifies to: $\frac{1}{2} \int_{0}^{\frac{\pi}{4}} \sec \theta , d\theta$

   • The second term simplifies to: $\frac{1}{2} \int_{0}^{\frac{\pi}{4}} \sec \theta \tan \theta , d\theta$

4. Integrate the Terms:

   • The integral of $\sec \theta$ is: $\int \sec \theta , d\theta = \ln |\sec \theta + \tan \theta|$

   • The integral of $\sec \theta \tan \theta$ is: $\int \sec \theta \tan \theta , d\theta = \sec \theta$

5. Apply the Limits:

   Now, apply the limits of integration from 0 to $\frac{\pi}{4}$:

   • For the first term: $\frac{1}{2} \left[\ln |\sec \theta + \tan \theta|\right]_{0}^{\frac{\pi}{4}}$

     Evaluate at the limits: $\sec \frac{\pi}{4} = \sqrt{2}, \quad \tan \frac{\pi}{4} = 1$ $\sec 0 = 1, \quad \tan 0 = 0$

     So: $\frac{1}{2} \left[\ln (\sqrt{2} + 1) - \ln (1 + 0)\right] = \frac{1}{2} \ln (\sqrt{2} + 1)$

   • For the second term: $\frac{1}{2} \left[\sec \theta\right]_{0}^{\frac{\pi}{4}}$

     This evaluates to: $\frac{1}{2} \left[\sqrt{2} - 1\right]$

6. Combine the Results:

   The final result is the sum of both terms: $\frac{1}{2} \ln (\sqrt{2} + 1) + \frac{1}{2} (\sqrt{2} - 1)$

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Key Formulas or Methods Used

• Integral of Secant: $\int \sec \theta , d\theta = \ln |\sec \theta + \tan \theta|$

• Integral of Secant Tangent: $\int \sec \theta \tan \theta , d\theta = \sec \theta$

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Summary of Steps

1. Simplify the integrand using trigonometric identities.
2. Break the integral into two separate parts.
3. Simplify each integral:
   • $\int \sec \theta , d\theta$
   • $\int \sec \theta \tan \theta , d\theta$
4. Apply the limits of integration from 0 to $\frac{\pi}{4}$.
5. Combine the results to get the final answer: $\frac{1}{2} \ln (\sqrt{2} + 1) + \frac{1}{2} (\sqrt{2} - 1)$