Guided Solution: Integrating $\int \sqrt{\tan(x)} \sec(x) \, dx$

Q: Evaluate the integral $\int \sqrt{\tan(x)} \sec(x) \, dx$

Background

Topic: Integration Techniques (Substitution)

This question tests your ability to use substitution and algebraic manipulation to evaluate an integral involving trigonometric and radical functions. Recognizing the structure of the integrand is key to choosing an effective substitution.

Key Terms and Formulas

• Substitution Method: If $u = g(x)$, then $du = g'(x) dx$ and $\int f(g(x))g'(x) dx = \int f(u) du$.

• Trigonometric Identities: $\sec(x) = \frac{1}{\cos(x)}$, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, and $\frac{d}{dx} \tan(x) = \sec^2(x)$.

Step-by-Step Guidance

1. Let $u = \tan(x)$. Then, compute $du$ in terms of $dx$.

2. Recall that $\frac{d}{dx} \tan(x) = \sec^2(x)$, so $du = \sec^2(x) dx$.

3. Rewrite $\sec(x) dx$ in terms of $du$ and $u$ using the relationship above. Notice that $\sec(x) dx = \frac{du}{\sec(x)}$.

4. Express the original integral in terms of $u$ and $du$. Substitute $\sqrt{\tan(x)}$ with $\sqrt{u}$ and $\sec(x) dx$ as found in the previous step.

5. Set up the new integral in terms of $u$ and simplify as much as possible. You should now have an integral involving $u$ and possibly a rational exponent.

Try solving on your own before revealing the answer!

Final Answer:

Let $u = \tan(x)$, so $du = \sec^2(x) dx$ and $\sec(x) dx = \frac{du}{\sec(x)}$.

Rewrite the integral:

$\int \sqrt{\tan(x)} \sec(x) dx = \int \sqrt{u} \sec(x) dx$

Since $\sec^2(x) dx = du$, $\sec(x) dx = \frac{du}{\sec(x)}$.

But $\sec(x) = \sqrt{1 + \tan^2(x)} = \sqrt{1 + u^2}$, so $\sec(x) dx = \frac{du}{\sqrt{1 + u^2}}$.

Substitute into the integral:

$\int \sqrt{u} \cdot \frac{du}{\sqrt{1 + u^2}} = \int \frac{u^{1/2}}{\sqrt{1 + u^2}} du$

Now, use the substitution $u = \tan(x)$ and integrate:

$\int \frac{u^{1/2}}{\sqrt{1 + u^2}} du$

Let $u = \tan(x)$, so $1 + u^2 = 1 + \tan^2(x) = \sec^2(x)$, so $\sqrt{1 + u^2} = \sec(x)$.

Therefore, $\int \sqrt{\tan(x)} \sec(x) dx = \frac{2}{3} \tan^{3/2}(x) + C$

Final Answer: $\boxed{\frac{2}{3} \tan^{3/2}(x) + C}$

This result comes from recognizing the substitution and simplifying the integral to a basic power rule.