Question

Evaluate the integral:

\[ \int \frac{4x \cos \left( \sqrt{4x^2 + 7} \right)}{\sqrt{4x^2 + 7}} \, dx \]

• A. \( \frac{1}{2} \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• B. \( \frac{7}{2} \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• C. \( \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• D. \( \frac{1}{4} \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• E. \( \frac{7}{4} \sin \left( \sqrt{4x^2 + 7} \right) + C \)

Hint:

When solving integrals involving composite functions, use substitution to simplify the expression, and remember to revert to the original variable at the end.

Answer 1

The Correct Option is C

We are given the integral: \[ I = \int \frac{4x \cos \left( \sqrt{4x^2 + 7} \right)}{\sqrt{4x^2 + 7}} \, dx \] Step 1: Use substitution
Let \( u = \sqrt{4x^2 + 7} \). Then: \[ \frac{du}{dx} = \frac{8x}{2\sqrt{4x^2 + 7}} = \frac{4x}{\sqrt{4x^2 + 7}} \] Thus, we have \( du = \frac{4x}{\sqrt{4x^2 + 7}} \, dx \), and the integral becomes: \[ I = \int \cos(u) \, du \] Step 2: Integrate
The integral of \( \cos(u) \) is \( \sin(u) \), so we have: \[ I = \sin(u) + C \] Step 3: Substitute back \( u \)
Now substitute \( u = \sqrt{4x^2 + 7} \) back into the equation: \[ I = \sin \left( \sqrt{4x^2 + 7} \right) + C \] Thus, the value of the integral is: \[ \sin \left( \sqrt{4x^2 + 7} \right) + C \] Thus, the correct answer is option (C), \( \sin \left( \sqrt{4x^2 + 7} \right) + C \).