Question

If \( \int \sec^2(7 - 4x) \, dx = \tan(7 - 4x) + C \), then the value of \( a \) is:

• A. -4
• B. -0.25
• C. 3
• D. 7

Hint:

When solving integrals involving \( \sec^2(x) \), use the substitution method and recognize the derivative of \( \tan(x) \) as \( \sec^2(x) \).

Answer 1

The Correct Option is D

We are given the integral: \[ \int \sec^2(7 - 4x) \, dx = \tan(7 - 4x) + C \] To solve this, we use the substitution method. Let: \[ u = 7 - 4x \quad \Rightarrow \quad du = -4 dx \quad \Rightarrow \quad dx = -\frac{1}{4} du \] Substituting this into the integral, we get: \[ \int \sec^2(u) \cdot \left(-\frac{1}{4}\right) \, du = -\frac{1}{4} \int \sec^2(u) \, du \] We know that: \[ \int \sec^2(u) \, du = \tan(u) \] Therefore, the integral becomes: \[ -\frac{1}{4} \cdot \tan(u) + C = -\frac{1}{4} \cdot \tan(7 - 4x) + C \] Thus, the result matches \( \tan(7 - 4x) + C \) when the value of \( a = 7 \). The correct answer is \( \boxed{7} \).