Evaluate int0infty x3 sqrt1-x4 dx | Mathematics I 2075

Mathematics I20755 marksNumerical: improper integral evaluationImproper integrals

Evaluate $\int_0^{\infty} x^3 \sqrt{1-x^4} dx$ [5]

Integrand: $f(x) = x^3\sqrt{1-x^4}$ - Limits: from $0$ to $\infty$ For the square root $\sqrt{1-x^4}$ to be real, we need: $1 - x^4 \geq 0 \implies x^4 \leq 1 \implies x \leq 1$ On $[0,\infty)$ this means the integrand is real only for $0 \leq x \leq 1$ . For $x 1$ , $1-x^4 < 0$ and the integra...

Integrand: $f(x) = x^3\sqrt{1-x^4}$ - Limits: from $0$ to $\infty$ For the square root $\sqrt{1-x^4}$ to be real, we need: $$1 - x^4 \geq 0 \implies x^4 \leq 1 \implies x \leq 1$$ On $[0,\infty)$ this means the integrand is real only for $0 \leq x \leq 1$. For $x 1$, $1-x^4 < 0$ and the integra...