Question

integraçao por partes x*√1-x*dx

Answer 1

$\int\ x*\sqrt{1-x}\ dx\\\\ u=x\rightarrow\ du=dx\\\\ \sqrt{1-x}=dv\rightarrow\ v=-\frac{2}{3}(1-x)^{\frac{3}{2}}$

Pela definição:

$\boxed{\int\ udv = uv-\int\ vdu}$

Resolvendo:

$\int\ x*\sqrt{1-x}\ dx = x*(-\frac{2}{3}(1-x)^{\frac{3}{2}})-(\int\ -\frac{2}{3}(1-x)^{\frac{3}{2}}\ dx)\\\\ \int=-\frac{2x}{3}(1-x)^{\frac{3}{2}}+\frac{2}{3}\int\ (1-x)^{\frac{3}{2}}\ dx\\\\ \boxed{\int=-\frac{2x}{3}(1-x)^{\frac{3}{2}}-\frac{4}{15}(1-x)^{\frac{5}{2}}+C}$

Answer 2

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Ola Higor

∫ x*√(x - 1) * dx

u = 1 - x e du = x

∫ (u^(3/2 - √u) du 

∫ u^(3/2) du - √u du 

= 2u^(5/2)/5  - 2u^(3/2) 

= 2/5 * (1 - x)^(5/2)- 2/3* (1 - x)^(3/2) 

= -2/15 * (1 - x)^(3/2) *(3x + 2) + C

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