Question

(x^3 dx)/(〖(5x〗^4+3)〖^5〗) решите интеграл

Answer 1

interpret ... (x^3 dx)/( ( [5x]^4+3) [^5] ) ...    $\Rightarrow \int{ \frac{ x^3 dx }{ ( (5x)^4 + 3 )^5 } } = \frac{1}{5^4} \int{ \frac{ (5x)^3 d (5x) }{ ( (5x)^4 + 3 )^5 } } =$

$= \frac{1}{ 4 \cdot 5^4 } \int{ \frac{1}{ ( (5x)^4 + 3 )^5 } } \, \cdot 4 \cdot (5x)^3 d (5x) = \frac{1}{ 4 \cdot 5^4 } \int{ \frac{1}{ ( (5x)^4 + 3 )^5 } } \, d (5x)^4 = \\\\ = \frac{1}{ 4 \cdot 5^4 } \int{ \frac{ d ( (5x)^4 + 3 ) }{ ( (5x)^4 + 3 )^5 } } = \frac{1}{ 4 \cdot 5^4 } \int{ ( (5x)^4 + 3 )^{-5} } \, d ( (5x)^4 + 3 ) = \\\\ = \frac{1}{ 1 + [- 5] } \cdot \frac{1}{ 4 \cdot 5^4 } \cdot ( (5x)^4 + 3 )^{-4} + C = \frac{1}{4} \cdot \frac{1}{ 4 \cdot 5^4 } \cdot \frac{1}{ ( (5x)^4 + 3 )^4 } + C = \\\\ = \frac{1}{ 2^2 \cdot 2^2 \cdot 5^4 } \cdot \frac{1}{ ( (5x)^4 + 3 )^4 } + C = \frac{1}{ 2^4 \cdot 5^4 } \cdot \frac{1}{ ( (5x)^4 + 3 )^4 } + C = \frac{1}{ 10^4 } \cdot \frac{1}{ ( (5x)^4 + 3 )^4 } + C \ ;$

О т в е т :    $\int{ \frac{ x^3 dx }{ ( (5x)^4 + 3 )^5 } } = \frac{1}{ 10 \ 000 \ ( (5x)^4 + 3 )^4 } + C \ .$