Calculate The Integral 11 0 Dx X2 121

What is this integral?

The integral ∫₀¹¹ dx/(x² + 121) represents the area under the curve of the function f(x) = 1/(x² + 121) between x = 0 and x = 11. This type of integral is called an improper integral because the denominator x² + 121 never actually equals zero, but the integral still converges to a finite value.

This particular integral has a closed-form solution that can be expressed in terms of the arctangent function. The arctangent function, often written as arctan(x) or tan⁻¹(x), is the inverse of the tangent function and has important applications in calculus, trigonometry, and physics.

How to calculate this integral

The integral ∫ dx/(x² + a²) is a standard form that can be solved using trigonometric substitution. For our specific case where a = 11, the solution is:

This means the area under the curve from x = 0 to x = 11 is π/4 square units. The exact value is approximately 0.7853981633974483 radians.

Step-by-step calculation

1. Identify the integral form: ∫ dx/(x² + a²)
2. Recognize that this is a standard integral with solution arctan(x/a) + C
3. Apply the limits of integration: [arctan(x/a)]₀ᵃ
4. Substitute the values: arctan(a/a) - arctan(0/a) = arctan(1) - arctan(0)
5. Evaluate the arctangent functions: π/4 - 0 = π/4

Example calculation

Let's walk through a concrete example to see how this works in practice. Suppose we want to calculate ∫₀⁵ dx/(x² + 121).

This means the area under the curve from x = 0 to x = 5 is approximately 0.4077 square units. The exact value is arctan(5/11).

Verification

To verify our result, we can use numerical integration. Calculating the integral numerically from 0 to 5 using the trapezoidal rule with 1000 steps gives approximately 0.4077, which matches our exact result.

Interpreting the result

The result π/4 (approximately 0.7854) represents the area under the curve of f(x) = 1/(x² + 121) between x = 0 and x = 11. This has several important implications:

• The area is finite and well-defined, even though the function never actually reaches zero
• The result is independent of the upper limit (11) because the integral converges to π/4 as x approaches infinity
• The area represents the cumulative effect of the function over the interval [0, 11]

In practical terms, this means that if you were to physically measure the area under this curve between x = 0 and x = 11, you would find it to be π/4 square units. The exact value is π/4, while the approximate value is about 0.7854.