Improper integrals
Problem 4.40 · medium
Evaluate \( \displaystyle \int_{1}^{\infty} \frac{1}{x^{4}} \, dx \), or show that it diverges.
- The integral is improper at infinity; replace that bound by T and take a limit.
- \[ \frac{d}{d x} \left(- \frac{1}{3 x^{3}}\right) = \frac{1}{x^{4}} \]An antiderivative, checked by differentiating.✓ Proved
- \[ \frac{1}{3} - \frac{1}{3 T^{3}} = \frac{T^{3} - 1}{3 T^{3}} \]The integral with T in place of the bad bound.✓ Proved
- As T → \infty, this approaches 1/3.
- \[ \lim_{T \to \infty}\left(\frac{T^{3} - 1}{3 T^{3}}\right) = \frac{1}{3} \]The limit.✓ Proved
Answer \( \frac{1}{3} \)
Lines: 3 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Not checked | — | a sentence; read, not computed |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | numerical quadrature over the whole interval agrees |
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/improper_integral, checked 2026-09-26 with SymPy 1.14.0.