find-x-3-1-x-2-dx-

Question Number 32269 by abdo imad last updated on 22/Mar/18

f i n d \int \frac{x^{3}}{\sqrt{1 + x^{2}}} d x

Answered by mrW2 last updated on 22/Mar/18

\int \frac{x^{3}}{\sqrt{1 + x^{2}}} d x

\frac{1}{2} \int \frac{x^{2}}{\sqrt{1 + x^{2}}} {d x}^{2}

\frac{1}{2} \int \frac{x^{2} + 1 - 1}{\sqrt{1 + x^{2}}} {d x}^{2}

\frac{1}{2} {\int \sqrt{x^{2} + 1} {d x}^{2} - \int \frac{1}{\sqrt{1 + x^{2}}} {d x}^{2}}

\frac{1}{2} {\int \sqrt{t + 1} d t - \int \frac{1}{\sqrt{1 + t}} d t}

\frac{1}{2} {\frac{2}{3} (t + 1) \sqrt{t + 1} - 2 \sqrt{t + 1}}

{\frac{1}{3} (t + 1) - 1} \sqrt{t + 1}

\frac{(t - 2) \sqrt{t + 1}}{3} + C

\frac{(x^{2} - 2) \sqrt{x^{2} + 1}}{3} + C

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