Evaluate the vector integral from $t=0$ to $t=4$ of the vector function $sec\left(t\right)tan\left(t\right)\mathbf{i}+tcos\left(2t\right)\mathbf{j}+{sin}^2\left(2t\right)cos\left(2t\right)\mathbf{k}$ with respect to $t$.

• A. $(tan(4)-tan(0\left)\right)\mathbf{i}+\left(\frac{1}{2}\right)sin\left(8\right)\mathbf{j}+\left(\frac{1}{6}\right){sin}^3\left(8\right)\mathbf{k}$
• B. $(tan(4)-tan(0\left)\right)\mathbf{i}+\left(\frac{1}{2}\right)sin\left(8\right)\mathbf{j}+\left(\frac{1}{6}\right){sin}^3\left(8\right)\mathbf{k}$
• C. $(tan(4)-tan(0\left)\right)\mathbf{i}+\left(8\right)\mathbf{j}+\left(\frac{1}{6}\right){sin}^3\left(8\right)\mathbf{k}$
• D. $(tan(4)-tan(0\left)\right)\mathbf{i}+\left(\frac{1}{2}\right)sin\left(8\right)\mathbf{j}+\left(\frac{1}{6}\right){sin}^3\left(8\right)\mathbf{k}$

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with evaluating a vector integral of the given vector function: sec(t) tan(t) i + t cos(2t) j + sin^2(2t) cos(2t) k with respect to t, over the interval [0, 4]. This involves integrating each component of the vector function separately.
2. Step 2: Break the vector function into its components. The vector function is composed of three components: (1) sec(t) tan(t) for the i component, (2) t cos(2t) for the j component, and (3) sin^2(2t) cos(2t) for the k component. Each of these will be integrated individually.
3. Step 3: Integrate the i component. The integral of sec(t) tan(t) with respect to t is sec(t), so evaluate sec(t) from t = 0 to t = 4. This will contribute to the i component of the result.
4. Step 4: Integrate the j component. The integral of t cos(2t) with respect to t requires integration by parts. Let u = t and dv = cos(2t) dt. Use the formula ∫u dv = uv - ∫v du to compute this integral, and evaluate the result from t = 0 to t = 4.
5. Step 5: Integrate the k component. The integral of sin^2(2t) cos(2t) with respect to t can be simplified using the substitution u = sin(2t), which makes du = 2 cos(2t) dt. Rewrite the integral in terms of u and evaluate it from t = 0 to t = 4. Combine the results of all three components to form the final vector.