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MathCalculus

Anti Derivative: Solve f′(x) = x⁻³ with f(1) = 1

Integrate x⁻³, determine the constant from f(1)=1, and verify the solution on the interval containing the initial point.

Question

Solve the initial value problem:

f′(x)=x−3,f(1)=1.f'(x)=x^{-3},\qquad f(1)=1.

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Evidence boundary

This page uses the derivative and initial value in OpenStax Calculus Volume 1, §4.10, exercise 499. The initial condition determines the solution on the interval containing x=1; it does not fix a constant on the negative half-axis. Both the derivative and initial value are checked explicitly.