Calculate The Given Integral Tan4 11x Dx

Calculating the integral of tan(4x) requires understanding trigonometric identities and integration techniques. This guide explains the process step-by-step, provides an interactive calculator, and includes common questions about solving this integral.

How to Calculate the Integral of tan(4x)

The integral of tan(4x) can be found using substitution and trigonometric identities. Here's a step-by-step breakdown:

Recognize that tan(4x) can be rewritten using the identity tan(θ) = sin(θ)/cos(θ).
Let u = 4x, then du = 4dx, which means dx = du/4.
Substitute into the integral: ∫tan(4x)dx = ∫(sin(4x)/cos(4x))dx = (1/4)∫(sin(u)/cos(u))du.
Recognize that sin(u)/cos(u) = tan(u), so the integral becomes (1/4)∫tan(u)du.
Use the identity ∫tan(u)du = -ln|cos(u)| + C.
Substitute back u = 4x: (1/4)[-ln|cos(4x)|] + C = -1/4 ln|cos(4x)| + C.

Note: The absolute value is used because the logarithm function is only defined for positive arguments.

Formula Used

The integral of tan(4x) with respect to x is given by:

∫tan(4x)dx = -1/4 ln|cos(4x)| + C

Where C is the constant of integration.

This formula is derived using the substitution method and fundamental trigonometric identities.

Worked Example

Let's calculate the definite integral of tan(4x) from 0 to π/8:

Using the formula: ∫[0 to π/8] tan(4x)dx = -1/4 [ln|cos(4x)|] evaluated from 0 to π/8.
At x = π/8: cos(4*(π/8)) = cos(π/2) = 0. The natural logarithm of 0 is undefined, so we must consider the limit as x approaches π/8 from the left.
As x approaches π/8, cos(4x) approaches 0, and ln|cos(4x)| approaches -∞.
At x = 0: cos(0) = 1, so ln|cos(0)| = ln(1) = 0.
The definite integral is -1/4 [lim(x→π/8⁻) ln|cos(4x)| - ln|cos(0)|].
Since the limit is -∞, the integral diverges to -∞.

This example shows that the integral of tan(4x) over certain intervals may diverge, requiring careful consideration of the limits.

Frequently Asked Questions

What is the integral of tan(4x)?

The integral of tan(4x) is -1/4 ln|cos(4x)| + C, where C is the constant of integration.

Can the integral of tan(4x) be evaluated over all real numbers?

No, the integral of tan(4x) over all real numbers diverges because the function has vertical asymptotes at x = π/8 + kπ/4 for any integer k.

What substitution is used to integrate tan(4x)?

The substitution u = 4x is used, which transforms the integral into (1/4)∫tan(u)du.

Is there a simpler form for the integral of tan(4x)?

The expression -1/4 ln|cos(4x)| + C is the simplest form of the integral of tan(4x).