Question

Evaluate the integral: $ \int \frac{\sec^2(\sqrt{2x+5})}{\sqrt{2x+5}} \, dx $

• A. \( \frac{1}{2} \log |2x + 5| \)
• B. \( \frac{1}{\sqrt{2x+5}} \)
• C. \( \frac{1}{2} \sec^2(\sqrt{2x+5}) \)
• D. \( \frac{1}{\sqrt{2x+5}} + C \)

Hint:

When solving integrals involving functions like \( \sec^2 \), consider using substitutions such as \( u = \sqrt{2x+5} \) to simplify the expression.

Answer 1

The Correct Option is A

We are given the integral: \[ \int \frac{\sec^2(\sqrt{2x+5})}{\sqrt{2x+5}} \, dx \] To solve this, let's substitute \( u = \sqrt{2x + 5} \). This implies: \[ du = \frac{1}{2\sqrt{2x + 5}} \cdot 2 \, dx = \frac{1}{\sqrt{2x+5}} \, dx \] 
Thus, the integral becomes: \[ \int \sec^2(u) \, du = \tan(u) + C \] Now, substituting \( u = \sqrt{2x+5} \), we get: \[ \tan(\sqrt{2x+5}) + C \] The correct answer is \( \frac{1}{2} \log |2x + 5| \).