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Tema: Integralas ∫(x²+1)1^2 * x³ sprendimas dalimis, keitimai?

Integralas ∫(x²+1)1^2 * x³

Nebuvau susipazines su sprendimu dalimis anksciau, bet sio vistiek nepavykta iveikti..:)

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  • Registravosi: 2011-06-08
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Ats: Integralas ∫(x²+1)1^2 * x³ sprendimas dalimis, keitimai?

\int{\sqrt{x^{2}+1} \cdot x^{3}}dx



x^{2}=u, \quad \quad \quad \quad \quad dx=u \prime du=(x^{2}) \prime du=2xdu

\int{\sqrt{u+1} \cdot ux}dx= \frac{1}{2}\int{\sqrt{u+1} \cdot u \cdot 2x}dx=\frac{1}{2}\int{\sqrt{u+1} \cdot u}du



u+1=s, \quad \quad \quad \quad \quad u=s-1, \quad \quad \quad \quad \quad du=s \prime ds = (u+1) \prime ds = 1 \cdot ds = ds

\frac{1}{2} \int{\sqrt{s} \cdot (s-1)}ds = \frac{1}{2}\int{(s^{\frac{3}{2}}-s^{\frac{1}{2}})}ds=
=\frac{1}{2}\cdot\frac{s^{\frac{5}{2}}}{\frac{5}{2}}-\frac{1}{2}\cdot\frac{s^{\frac{3}{2}}}{\frac{3}{2}}+C=\frac{s^{\frac{5}{2}}}{5}-\frac{s^{\frac{3}{2}}}{3}+C


s=u+1, \quad \quad \quad \quad \quad u=x^{2}
\frac{(x^{2}+1)^{\frac{5}{2}}}{5}-\frac{(x^{2}+1)^{\frac{3}{2}}}{3}+C=\frac{\sqrt{(x^{2}+1)^{5}}}{5}-\frac{\sqrt{(x^{2}+1)^{3}}}{3}+C

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