Evaluate \(\int_0^∞ x^3 \sqrt{1 – x^4}\) dx

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Solution:

\(\int_0^∞ x^3 \sqrt{1 – x^4}\)

\( = \lim_{h \to ∞} \int_0^h x ^ 3 \sqrt{1-x^4} dx\)   ….. (i)

Take

\(\int \sqrt{1-x^4} dx\)

Put 1 – x4 = u then -4x3 dx = du

\(= – \frac{1}{4} \int \sqrt{u} du\)

\(= – \frac{1}{4} \times \frac{2}{3} u^{\frac{3}{2}}\)

\(= – \frac{1}{6} ( 1 – x^4 )^{\frac{3}{2}}\)

Thus

\(\lim_{h \to ∞} \int_0^h x ^ 3 \sqrt{1-x^4} dx = \lim_{h \to ∞} \left [ -\frac{1}{6} (1 – x^4)^{\frac{3}{2}} \right ]_0^h\)

\(= \lim_{h \to ∞} -\frac{1}{6} \left [ (1 – h^4)^{\frac{3}{2}} – 1 \right ]\)

\(= -\frac{1}{6} \left [ (1 – ∞)^{\frac{3}{2}} – 1 \right ]\)

\(= -\frac{1}{6} \times ∞\)

= ∞

 

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