Question

(ax²+bx+c)√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.

The solution is simple and easy to understand.

See the attachment for detailed solution.

Answer 2

we have to find the value of $\int{(ax^2+bx+c)\sqrt{x}}\,dx$

=$\int{ax^2\sqrt{x}+bx\sqrt{x}+c\sqrt{x}}\,dx$

=$\int{ax^{5/2}+bx^{3/2}+cx^{1/2}}\,dx$

=$\int{ax^{5/2}}\,dx+\int{bx^{3/2}}\,dx+\int{cx^{1/2}}\,dx$

=$a\int{x^{5/2}}\,dx+b\int{x^{3/2}}\,dx+c\int{x^{1/2}}\,dx$

=$\frac{2a}{7}x^{7/2}+\frac{2b}{5}x^{5/2}+\frac{2c}{3}x^{3/2}$