∫

Calculus

University

Calculus (University Level)

Improper Integral (1/xᵖ, 1 to ∞)

Determine convergence and value of ∫[1,∞] x^(−p) dx.

Standard Formula

∫[1,∞] 1/xᵖ dx = 1/(p−1) if p > 1, else diverges

Interactive Sandbox

Live Reactive

Adjust values in real-time and observe mathematical cause-and-effect

Exponent p (must be > 0)(Real number)

120

Live EquationFormula Proof

∫[1,∞] 1/xᵖ dx = 1/(2−1) if 2 > 1, else diverges = 1

Area Under Curve ∫f(x)dx

Calculated Solution

Direct computational model evaluating Improper Integral (1/xᵖ, 1 to ∞).

Variable Mechanics

Roles, constraints & impacts

1 Params

Exponent p (must be > 0)

ImpactPrimary formula parameter

ConstraintReal number

At a Glance

Gotchas, units & real-world use

Essential

Formula Nature & Mechanics

Direct computational model evaluating Improper Integral (1/xᵖ, 1 to ∞).

Common Mistakes & Traps

Ensure all input measurements share consistent units before calculating.
Check for denominator values approaching zero to prevent undefined division.
Follow standard order of operations (PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract).

Dimensional Analysis & Units

Direct evaluation matching formula variable dimensions.

Real-World & Industry Application

Widely utilized across modern calculus applications, academic analysis, and engineering estimations.

📖 Beginner's guideImproper Integrals: Convergence and the p-TestLearn what an improper integral is, when ∫1/xᵖ from 1 to ∞ converges, and how to find its value, with a free calculator with step-by-step solutions.

Worked Example: Improper Integral (1/xᵖ, 1 to ∞)

Given:

p = 2

∫[1,∞] 1/x² dx
= [−1/x]₁^∞ = 0 − (−1) = 1

Answer: 1

Frequently Asked Questions

What is the Improper Integral (1/xᵖ, 1 to ∞) formula?

∫[1,∞] 1/xᵖ dx = 1/(p−1) if p > 1, else diverges. Determine convergence and value of ∫[1,∞] x^(−p) dx.

How do I calculate Improper Integral (1/xᵖ, 1 to ∞)?

Enter Exponent p (must be > 0) (p) into the calculator. It applies ∫[1,∞] 1/xᵖ dx = 1/(p−1) if p > 1, else diverges and shows every step of the working.

Related Calculus Calculators

Power Rule for Derivativesd/dx(xⁿ) = n × x^(n-1)Definite Integral (Power Rule)∫[a to b] x^n dx = [x^(n+1)/(n+1)]|[a to b]Partial Derivative∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/h Chain Rule (Derivative)d/dx[f(g(x))] = f'(g(x)) · g'(x)Product Rule (Derivative)d/dx[f·g] = f'·g + f·g'Quotient Rule (Derivative)d/dx[f/g] = (f'·g − f·g') / g²

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