3.3 Q-7

Question Statement

Evaluate the following integral:

$$\int \frac{\sec^2 x}{\sqrt{\tan x}} , dx$$

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Background and Explanation

This integral involves trigonometric functions, specifically secant and tangent. To solve, we need to recognize a substitution that simplifies the expression. The square root of tangent and the secant squared suggest the use of a substitution to convert the integral into a more manageable form.

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Solution

1. Rewrite the integral:
   Start by recognizing the structure of the integrand. We express the integral as:

$$\int \frac{\sec^2 x}{\sqrt{\tan x}} , dx$$

We can rewrite it as:

$$\int (\tan x)^{\frac{1}{2}} \cdot \sec^2 x , dx$$

2. Use substitution:
   Now, notice that the derivative of $\tan x$ is $\sec^2 x$. We can let:

$$u = \tan x$$

Then, the derivative of $u$ is:

$$du = \sec^2 x , dx$$

Substituting this into the integral:

$$= \int u^{\frac{1}{2}} , du$$

3. Integrate:
   Use the power rule of integration, which states:

$$\int u^n , du = \frac{u^{n+1}}{n+1}$$

Applying this rule to our integral, where $n = \frac{1}{2}$:

$$= \frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C$$

Simplify the exponent:

$$= \frac{u^{\frac{3}{2}}}{\frac{3}{2}} + C$$

This simplifies further to:

$$= \frac{2}{3} u^{\frac{3}{2}} + C$$

4. Substitute back:
   Recall that $u = \tan x$, so:

$$= \frac{2}{3} (\tan x)^{\frac{3}{2}} + C$$

Alternatively, we can express this as:

$$= 2 \sqrt{\tan x} + C$$

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Key Formulas or Methods Used

• Substitution:
  We substituted $u = \tan x$ to simplify the integral.

• Power Rule of Integration:
  We applied the power rule to integrate expressions of the form $u^n$.

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Summary of Steps

1. Rewrite the integral as $\int (\tan x)^{\frac{1}{2}} \sec^2 x , dx$.
2. Use substitution: $u = \tan x$ and $du = \sec^2 x , dx$.
3. Integrate using the power rule.
4. Substitute back $u = \tan x$ to obtain the final result.