Question

$\frac{\int\ {x^{2}+5x-1 } \, }{\sqrt{x} } dx$ Please solve this one guys, its urgent!

Answer 1

Here, We integrated all the terms step by step.

And at last we used a formula

i.e.

Integration of xⁿ = xⁿ+¹/n+1

Answer 2

ANSWER:

$\sf{\frac{2}{5} \big( {x}^{ \frac{5}{2} } \big) + \frac{10}{3} \big( {x}^{ \frac{3}{2} } \big) - 2 \sqrt{x} + C}$

GIVEN:

$\displaystyle\int\frac{ {x^{2}+5x-1 } }{\sqrt{x} } dx$

TO INREGRATE:

$\frac{ {x^{2}+5x-1 } }{\sqrt{x} } dx$

FORMULAE:

$\displaystyle \frac{ {x}^{m} }{ {x}^{n} } = {x}^{m - n} \\ \\ \displaystyle \int {x}^{n} = \frac{ {x}^{n + 1} }{n + 1} + C$

EXPLANATION:

$\displaystyle \int\frac{ {x^{2}+5x-1} }{\sqrt{x} } dx \\ \\ \\ \textsf{let x = t} \\ \\ \\ \int\frac{ {t^{2}+5t-1} }{\sqrt{t} } dx \\ \\ \\ \int \bigg(\frac{ {t}^{2} }{ \sqrt{t} } \bigg) dx + \int\bigg( \frac{5t}{ \sqrt{t} } \bigg)dx - \int\bigg( \frac{1}{ \sqrt{t} } \bigg)dx \\ \\ \\ \int {t}^{(2 - \frac{1}{2} )} dx + 5\int {t}^{(1 - \frac{1}{2} )} dx - \int {t}^{( - \frac{1}{2} )} dx \\ \\ \\ \int {t}^{( \frac{3}{2} )} dx + 5\int {t}^{( \frac{1}{2} )} dx - \int {t}^{( - \frac{1}{2} )} dx \\ \\ \\ \frac{ {t}^{( \frac{3}{2} + 1) } }{ \frac{3}{2} + 1} + C_1 + 5 \bigg(\frac{ {t}^{( \frac{1}{2} + 1) } }{ \frac{1}{2} + 1} \bigg) + C_2 - {t}^{ (- \frac{1}{2} + 1 )} - C_3 \\ \\ \\ \frac{ {t}^{ \frac{5}{2} } }{ \frac{5}{2} } + 5 \bigg(\frac{ {t}^{ \frac{3}{2} } }{ \frac{3}{2} } \bigg) - \frac{ {t}^{ \frac{ 1}{2} } }{ \frac{ 1}{2} } + C_1 + C_2 - C_3 \\ \\ \\\textsf{Substitute t = x} \\ \\ \frac{2}{5} \big( {x}^{ \frac{5}{2} } \big) + \frac{10}{3} \big( {x}^{ \frac{3}{2} } \big) - 2 \sqrt{x} + C$

$\textsf{HERE C IS A CONSTANT OF INTEGRATION } \\ \textsf{AND IS EQUAL TO C_1 + C_2 - C_3 }$