3.6 Q-26

Question Statement

Evaluate the following definite integral:

$$\int_0^{\frac{\pi}{4}} \frac{\sin x - 1}{\cos^2 x} , dx$$

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Background and Explanation

This problem involves integrating a rational function with trigonometric expressions. To solve it, we use trigonometric identities and the standard integration formulas for secant and tangent functions. The key concepts required are:

• Secant and Tangent identities: $\sec^2 x = 1 + \tan^2 x$, and the integration formulas:
  • $\int \sec^2 x , dx = \tan x$
  • $\int \sec x \tan x , dx = \sec x$

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Solution

Step 1: Split the integrand

We start by rewriting the given integral:

$$\int_0^{\frac{\pi}{4}} \frac{\sin x - 1}{\cos^2 x} , dx$$

This can be split into two terms:

$$\int_0^{\frac{\pi}{4}} \left( \frac{\sin x}{\cos^2 x} - \frac{1}{\cos^2 x} \right) , dx$$

This gives us:

$$\int_0^{\frac{\pi}{4}} \sec x \tan x , dx - \int_0^{\frac{\pi}{4}} \sec^2 x , dx$$

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Step 2: Integrate each term

• First integral: The integral of $\sec x \tan x$ is straightforward:

$$\int \sec x \tan x , dx = \sec x$$

Evaluating from 0 to $\frac{\pi}{4}$:

$$\sec\left( \frac{\pi}{4} \right) - \sec(0)$$

Since $\sec\left( \frac{\pi}{4} \right) = \sqrt{2}$ and $\sec(0) = 1$, this gives:

$$\sqrt{2} - 1$$

• Second integral: The integral of $\sec^2 x$ is:

$$\int \sec^2 x , dx = \tan x$$

Evaluating from 0 to $\frac{\pi}{4}$:

$$\tan\left( \frac{\pi}{4} \right) - \tan(0)$$

Since $\tan\left( \frac{\pi}{4} \right) = 1$ and $\tan(0) = 0$, this gives:

$$1 - 0 = 1$$

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Step 3: Combine the results

Now, we combine the results of the two integrals:

$$\left( \sqrt{2} - 1 \right) - 1 = \sqrt{2} - 2$$

Thus, the value of the integral is:

$$\boxed{\sqrt{2} - 2}$$

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Key Formulas or Methods Used

• Integration of secant and tangent:
  • $\int \sec x \tan x , dx = \sec x$
  • $\int \sec^2 x , dx = \tan x$

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Summary of Steps

1. Split the integrand into two simpler terms:

$$\int_0^{\frac{\pi}{4}} \sec x \tan x , dx - \int_0^{\frac{\pi}{4}} \sec^2 x , dx$$

2. Integrate each term:

   • $\int \sec x \tan x , dx = \sec x$
   • $\int \sec^2 x , dx = \tan x$

3. Evaluate the integrals at the limits $0$ and $\frac{\pi}{4}$.

4. Combine the results to get the final answer: $\sqrt{2} - 2$

The final answer is:

$$\boxed{\sqrt{2} - 2}$$