Question

integration √x(x^2-5x+3)dx​

Answer 1

Explanation:

please refer to the attachment

hope it may help

Answer 2

Answer:-

We have to evaluate:

∫ √x(x² - 5x + 3) dx

using $\sf \sqrt[n]{x} = x^{\frac{1}{n}}$ we get,

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

using aᵐ × aⁿ = aᵐ⁺ⁿ we get,

$\implies \int \sf \:( {x}^{ \frac{1}{2} + 2 } - 5 \times {x}^{ \frac{1}{2} + 1} + 3 \times {x}^{ \frac{1}{2} } ) \: \: dx \\ \\ \\ \implies \int \sf \: ( {x}^{ \frac{1 + 4}{2}} - 5 \times {x}^{ \frac{1 + 2}{2}} + 3 \times {x}^{ \frac{1}{2} } ) \: \: dx \\ \\ \\ \implies \int \sf \: ({x}^{ \frac{5}{2} } - 5 {x}^{ \frac{3}{2} } + 3 {x}^{ \frac{1}{2} } ) \: \: dx$

using ∫ (u ± v) dx = ∫ u dx ± ∫ v dx we get,

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

using ∫ xⁿ dx = (xⁿ⁺¹ / n + 1) + c we get,

$\: \implies \sf \: \frac{ {x}^{ \frac{5}{2} + 1} }{ \frac{5}{2} + 1 } - 5 \times \bigg( \frac{ {x}^{ \frac{3}{2} + 1} }{ \frac{3}{2} + 1} \bigg) + 3 \times \bigg( \frac{ {x}^{ \frac{1}{2} + 1} }{ \frac{1}{2} + 1 } \bigg) + c \\ \\ \\ \implies \sf \: \frac{ {x}^{ \frac{5 + 2}{2} } }{ \frac{5 + 2}{2} } - 5 \times \frac{ {x}^{ \frac{3 + 2}{2} } }{ \frac{3 + 2}{2} } + 3 \times \frac{ {x}^{ \frac{1 + 2}{2} } }{ \frac{1 + 2}{2} } + c \\ \\ \\ \implies \sf \: \frac{ {x}^{ \frac{7}{2} } }{ \frac{7}{2} } - 5 \times \frac{ {x}^{ \frac{5}{2} } }{ \frac{5}{2} } + 3 \times \frac{ {x}^{ \frac{3}{2} } }{ \frac{3}{2} } + c \\ \\ \\ \implies \sf \: \frac{2 {x}^{ \frac{7}{2} }}{7} - 5 \times \frac{2}{5} \times {x}^{ \frac{5}{2} } + 3 \times \frac{2}{3} \times {x}^{ \frac{3}{2} } + c \\ \\ \\ \implies \underline{ \underline{\sf \: \frac{2}{7} {x}^{ \frac{7}{2} } - 2 {x}^{ \frac{5}{2} } + 2 {x}^{ \frac{3}{2} } + c}}$