Question

Integral por método de substituição

Answer 1

∫$x^{3} (1+2 x^{4} )^{ \frac{3}{2} }$ dx

fazendo a substituição $u = 1+2 x^{4}$ temos:

$du = 8 x^{3} dx$
$\frac{du}{8} = x^{3} dx$

então:

∫$u^{ \frac{3}{2} } \frac{du}{8}$
$\frac{1}{8}$∫$u^{ \frac{3}{2} }$ du
$\frac{1}{8} \frac{ u^{ \frac{5}{2} } }{ \frac{5}{2} } + c$
$\frac{1}{20} (1+2 x^{4} )^{ \frac{5}{2} } + c$

Answer 2

$\int({x}^{3}{(1+2{x}^{4})}^{\frac{3}{2}})dx=$

$\frac{1}{8}\int(8{x}^{3}{(1+2{x}^{4})}^{\frac{3}{2}})dx$

faça

$u=1+2{x}^{4}\\du=8{x}^{3}dx$

$\frac{1}{8}\int(8{x}^{3}{(1+2{x}^{4})}^{\frac{3}{2}})dx=\frac{1}{8}\int({u}^{\frac{3}{2}})du$

$\frac{1}{8}\int({u}^{\frac{3}{2}})du=\frac{1}{20}.{u}^{\frac{5}{2}}+k$

$\int({x}^{3}{(1+2{x}^{4})}^{\frac{3}{2}})dx=\frac{1}{20}{(1+2{x}^{4})}^{\frac{5}{2}}+k$