Question

5x³+x²+2/√x ,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

In the attachment I have answered this problem.   I have applied decomposition method to find the anti derivative of the given function.   See the attachment for detailed solution.

Answer 2

Dear Student,

Answer:$\int{5 {x}^{3} + {x}^{2} + \frac{2}{ \sqrt{x}}}dx = \frac{5}{4} {x}^{4} + \frac{ {x}^{3} }{3} + 4\sqrt{x} + C$

Solution:

As we know that power rule of Integration is
$\int{{x}^{n}}dx = \frac{ {x}^{n + 1} }{n + 1} + C\: \: \: \:$

provided that n ≠ -1

So, here

$\int{5 {x}^{3}} dx +\int{{x}^{2}} dx + \int{2 \frac{1}{ \sqrt{x}}} dx \\ \\ = 5 \frac{ {x}^{3 + 1} }{3 + 1} + \frac{ {x}^{2 + 1} }{2 + 1} + 2 \frac{ {x}^{ \frac{ - 1}{2} + 1} }{ \frac{ - 1}{2} + 1} + C \\ \\ = \frac{5}{4} {x}^{4} + \frac{ {x}^{3} }{3} + 4 \sqrt{x} + C$

is the final answer.

Hope it helps you.