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计算$\int_0^1\sqrt{x(1+x)}\,\mathrm dx$

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isee Posted at 2022-12-13 13:26:48 |Read mode
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: $\int_0^1\sqrt{x(1+x)}\,\mathrm dx$ 怎么计算呢?




试双曲函数换元.

设 $x=\sinh^2 t$,则 \[\mathrm dx=2\sinh x\cosh x\,\mathrm dt=\sinh^2 2t\,\mathrm dt,\] 则\begin{align*}
\int_0^1\sqrt{x(1+x)}\mathrm dx&=\int_0^\mathrm {arcsinh\, 1} \sinh t\cosh t\cdot \sinh 2t\ \mathrm dt\\[1em]
&=\frac 12\int_0^\mathrm {arcsinh\, 1} \sinh^2 2t\ \mathrm dt\\[1em]
&=\frac 12\int_0^\mathrm {arcsinh\, 1} \frac {\cosh 4t-1}2\ \mathrm dt\\[1em]
&=\frac 14\int_0^\mathrm {arcsinh\, 1} \cosh 4t\ \mathrm dt-\frac 14\int_0^\mathrm {arcsinh\, 1} \ \mathrm dt\\[1em]
&=\frac 1{16}\sinh 4t-\frac 14t\bigg|_0^\mathrm {arcsinh\, 1}\\[1em]
&=\frac 14\sinh t\cosh t\cosh 2t-\frac 14t\bigg|_0^\mathrm {arcsinh\, 1}\\[1em]
&=\frac {3\sqrt 2}4-\frac 14\ln( 1+\sqrt 2).
\end{align*}

isee=freeMaths@知乎

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2025-4-21 19:17 GMT+8

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