4sec 2 X 16 Trig Integral Calculator

This calculator computes the integral of 4sec²(2x) × 16. The result is expressed in terms of the natural logarithm function, providing both the exact form and a decimal approximation.

How to Use This Calculator

To calculate the integral of 4sec²(2x) × 16:

Enter the lower limit of integration (a)
Enter the upper limit of integration (b)
Click "Calculate" to compute the result
Review the exact form and decimal approximation

The calculator uses the standard trigonometric integral formula and provides both exact and approximate results for your convenience.

Formula Explained

The integral of 4sec²(2x) × 16 is calculated using the following steps:

Step 1: Simplify the Expression

First, simplify the integrand:

4sec²(2x) × 16 = 64sec²(2x)

Step 2: Apply the Integral Formula

The integral of sec²(u) is tan(u) + C. Here, u = 2x:

∫64sec²(2x) dx = 64 × (1/2)tan(2x) + C = 32tan(2x) + C

Step 3: Apply Limits of Integration

Evaluate the antiderivative at the upper and lower limits:

∫[a to b] 64sec²(2x) dx = 32[tan(2b) - tan(2a)]

This gives the exact form of the integral. The calculator also provides a decimal approximation for practical use.

Worked Example

Let's calculate the integral from 0 to π/4:

Lower limit (a) = 0
Upper limit (b) = π/4
Exact form: 32[tan(π/2) - tan(0)] = 32[∞ - 0] = ∞
Decimal approximation: The integral diverges to infinity

Note

This example shows that the integral of 4sec²(2x) × 16 from 0 to π/4 diverges to infinity. The calculator will indicate this behavior when the limits include points where the tangent function is undefined.

Frequently Asked Questions

What is the integral of sec²(x)?: The integral of sec²(x) is tan(x) + C, where C is the constant of integration.
How do I handle limits where tan(2x) is undefined?: The calculator will indicate when the integral diverges to infinity at points where tan(2x) is undefined.
Can I use this calculator for other trigonometric integrals?: This calculator is specifically designed for integrals involving 4sec²(2x) × 16. For other trigonometric integrals, please use our dedicated calculators.
What units should I use for the limits of integration?: The calculator expects the limits to be in radians. If you need degrees, convert them to radians first.