Improper Integrals

Improper Integrals

An integral with an infinite limit is defined as a limit:

$int_a^{infty} f(x) , dx = lim_{T o infty} int_a^T f(x) , dx$

If the limit exists, the integral converges. Otherwise, it diverges.

Key Examples

The p-Test

int_1^{infty} rac{1}{x^p} , dx converges if and only if $p > 1$.

This is why the harmonic series sum rac{1}{n} diverges ($p=1$), but sum rac{1}{n^2} converges ($p=2$).

Numerical Approach

Integrate to a large finite $T$:

$int_a^{infty} f(x) , dx approx int_a^T f(x) , dx quad (T ext{ large})$

The truncation error is roughly $left|int_T^{infty} f(x) , dx ight|$. For rac{1}{x^2}, this is rac{1}{T}, so $T=100000$ gives 5-decimal accuracy.

Your Task

Implement double improper_integral(double (*f)(double), double a, double T, int n) that approximates $int_a^T f(x) , dx$ using the midpoint rule.