Calculate Integral Tan 1 X Chegg

The integral of tan(1/x) is a fundamental calculus problem that appears in various mathematical contexts. This guide explains how to compute it, provides an interactive calculator, and offers practical examples.

What is the integral of tan(1/x)?

The integral of tan(1/x) with respect to x is a definite integral that represents the area under the curve of the tangent function with argument 1/x. Mathematically, it's expressed as:

∫ tan(1/x) dx = -ln|cos(1/x)| + C

Where C is the constant of integration. This result comes from recognizing that tan(1/x) is the derivative of -ln|cos(1/x)|, and applying the fundamental theorem of calculus.

How to calculate the integral of tan(1/x)

To compute the integral of tan(1/x), follow these steps:

Recognize that tan(1/x) is the derivative of -ln|cos(1/x)|.
Apply the integral sign to both sides of the equation.
Combine the constants and add the constant of integration.

Note: The absolute value is necessary because the cosine function can be negative, and the logarithm is only defined for positive arguments.

Example calculation

Let's compute the definite integral from x=1 to x=2:

∫[1 to 2] tan(1/x) dx = -ln|cos(1/2)| - (-ln|cos(1/1)|)

= -ln(cos(0.5)) + ln(cos(1))

Using approximate values:

cos(0.5) ≈ 0.8776
cos(1) ≈ 0.5403

The result is approximately -ln(0.8776) + ln(0.5403) ≈ -0.1336 + (-0.6162) ≈ -0.7498.

Common pitfalls

When working with the integral of tan(1/x), be aware of these common mistakes:

Forgetting the absolute value in the logarithm: tan(1/x) can be negative, so the argument of the logarithm must be positive.
Omitting the constant of integration: this is essential for indefinite integrals.
Incorrectly applying the chain rule: remember that the derivative of 1/x is -1/x².

FAQ

What is the antiderivative of tan(1/x)?

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

Can I integrate tan(1/x) using substitution?

Yes, you can use substitution with u = 1/x, but it's simpler to recognize that tan(1/x) is the derivative of -ln|cos(1/x)|.

What's the domain of tan(1/x)?

The function tan(1/x) is undefined where cos(1/x) = 0, which occurs when 1/x = π/2 + kπ (k integer).