Question

an easy question from integration ​

Answer 1

Question

$\rm: \implies \: \int \dfrac{(1 + x) ^{3} }{ \sqrt{x} } \: \: dx$

Solution:-

$\rm: \implies \: \int \dfrac{(1 + x) ^{3} }{ \sqrt{x} } \: \: dx$

Using this identity

$\rm(a + b) {}^{3} = { a}^{3} + {b}^{3} + 3ab(a + b)$

We get

$\rm : \implies \: \int \dfrac{1 + {x}^{3} + 3 \times 1 \times x(1 + x) }{ {x}^{ \frac{1}{2} } } \: \: dx$

$\rm : \implies \: \int \dfrac{(1 + 3x + 3 {x}^{2} + {x}^{3} )}{ {x}^{ \frac{1}{2} } } \: \: dx$

Now split the integration with x

$\rm : \implies \: \int \dfrac{1}{x {}^{ \frac{1}{2} } } dx + 3 \int \dfrac{x}{ {x}^{ \frac{1}{2} } } dx + 3 \int \dfrac{ {x}^{2} }{ {x}^{ \frac{1}{2} } } dx + \int \dfrac{ {x}^{3} }{ {x}^{ \frac{1}{2} } } dx$

$\rm: \implies \: \int {x}^{ \frac{ - 1}{2} } dx + 3 \int {x}^{ \frac{1}{2} } dx + 3 \int {x}^{ \frac{3}{2} } dx + \int {x}^{ \frac{5}{2} } dx$

$\rm : \implies \dfrac{ {x}^{ \frac{ - 1}{2} + 1} }{ \frac{ - 1}{2} + 1} + \dfrac{ {3x}^{ \frac{1}{2} + 1} }{ \frac{1}{2} + 1} + \dfrac{ {3x}^{ \frac{3}{2} + 1} }{ \frac{3}{2} + 1} + \dfrac{ {x}^{ \frac{5}{2} + 1 } }{ \frac{5}{2} + 1 } + c$

$\rm : \implies \dfrac{ {x}^{ \frac{ 1}{2} } }{ \frac{ 1}{2} } + \dfrac{ {3x}^{ \frac{3}{2} } }{ \frac{3}{2} } + \dfrac{ {3x}^{ \frac{5}{2} } }{ \frac{5}{2} } + \dfrac{ {x}^{ \frac{7}{2} } }{ \frac{7}{2} } + c$

$\rm : \implies {2x}^{ \frac{1}{2} } + {2x}^{ \frac{3}{2} } + \dfrac{6}{5} {x}^{ \frac{5}{2} } + \dfrac{2}{7} {x}^{ \frac{7}{2} } + c$

Answer:-

$\rm : \implies \boxed{ \rm{2x}^{ \frac{1}{2} } + {2x}^{ \frac{3}{2} } + \dfrac{6}{5} {x}^{ \frac{5}{2} } + \dfrac{2}{7} {x}^{ \frac{7}{2} } + c}$