Find-x-5-x-3-1-dx-Answer-I-2-45-3x-3-2-x-3-1-3-c-

Question Number 179175 by Acem last updated on 26/Oct/22

F i n d \int x^{5} \sqrt{x^{3} + 1} d x

A n s w e r : I = \frac{2}{45} (3 x^{3} - 2) \sqrt{{(x^{3} + 1)}^{3}} + c

Commented by Acem last updated on 26/Oct/22

W o n d e r f u l S i r!

Commented by cortano1 last updated on 26/Oct/22

I = \int x^{3} (x^{2} \sqrt{x^{3} + 1}) dx

let \sqrt{x^{3} + 1} = h \Rightarrow x^{3} = h^{2} - 1

x^{2} dx = \frac{2}{3} h dh

I = \frac{2}{3} \int (h^{2} - 1) h^{2} dh = \frac{2}{3} \int (h^{4} - h^{2}) dh

= \frac{2}{3} (\frac{1}{5} h^{5} - \frac{1}{3} h^{3}) + c

= \frac{2}{3} h^{3} (\frac{1}{5} h^{2} - \frac{1}{3}) + c

= \frac{2}{3} \sqrt{{(x^{3} + 1)}^{3}} (\frac{1}{5} (x^{3} + 1) - \frac{1}{3}) + c

= \frac{2}{3} \sqrt{{(x^{3} + 1)}^{3}} (\frac{{3 x}^{3} - 2}{15}) + c

= \frac{2}{45} \sqrt{{(x^{3} + 1)}^{3}} ({3 x}^{3} - 2) + c