Calculate The Given Integral Tan4 6x Dx

How to calculate the integral of tan(4x)

Calculating the integral of tan(4x) with respect to 6x dx involves several steps. Here's a step-by-step guide:

1. First, rewrite the integral in terms of a single variable: ∫tan(4x) dx = (1/6)∫tan(4x) d(6x)
2. Use the substitution method: let u = 4x, then du = 4 dx
3. Adjust the integral: ∫tan(u) du = (1/6)∫tan(u) (du/4)
4. Simplify the expression: (1/24)∫tan(u) du
5. Integrate tan(u): (1/24)[-ln|cos(u)|] + C
6. Substitute back u = 4x: (1/24)[-ln|cos(4x)|] + C

Formula used

This formula is derived using substitution and the standard integral of tan(u).

Worked example

Let's calculate the integral from 0 to π/8:

1. Apply the formula: [(1/24)[-ln|cos(4x)|]] from 0 to π/8
2. Evaluate at π/8: (1/24)[-ln|cos(π/2)|] = (1/24)[-ln(0)] → undefined
3. This indicates a vertical asymptote at x = π/8, which is a point of discontinuity
4. The integral is improper and requires careful evaluation

Interpreting the result

The result of the integral is -1/24 * ln|cos(4x)| + C. This means:

• The integral represents the area under the curve of tan(4x)
• The result grows without bound as x approaches π/8 from the left
• The constant C represents the family of curves that differ by a vertical shift

In practical terms, this integral is useful in physics and engineering for analyzing oscillatory systems.