# What is int 4 x^5 -7 x^4 + 3 x^3 -8 x^2 -4 x + 1 dx?

Apr 13, 2018

$\implies \frac{2}{3} {x}^{6} - \frac{7}{5} {x}^{5} + \frac{3}{4} {x}^{4} - \frac{8}{3} {x}^{3} - 2 {x}^{2} + x + C$

#### Explanation:

$\int \left(4 {x}^{5} - 7 {x}^{4} + 3 {x}^{3} - 8 {x}^{2} - 4 x + 1\right) \mathrm{dx}$

$= 4 \int {x}^{5} \mathrm{dx} - 7 \int {x}^{4} \mathrm{dx} + 3 \int {x}^{3} \mathrm{dx} - 8 \int {x}^{2} \mathrm{dx} - 4 \int x \mathrm{dx} + \int \mathrm{dx}$

$= 4 \left({x}^{6} / 6\right) - 7 \left({x}^{5} / 5\right) + 3 \left({x}^{4} / 4\right) - 8 \left({x}^{3} / 3\right) - 4 \left({x}^{2} / 2\right) + x + C$

where $C$ is an arbitrary integration constant.

$= \frac{2}{3} {x}^{6} - \frac{7}{5} {x}^{5} + \frac{3}{4} {x}^{4} - \frac{8}{3} {x}^{3} - 2 {x}^{2} + x + C$

Apr 13, 2018

$\frac{2}{3} {x}^{6} - \frac{7}{5} {x}^{5} + \frac{3}{4} {x}^{4} - \frac{8}{3} {x}^{3} - 2 {x}^{2} + x + c$

#### Explanation:

$\text{integrate each term using the "color(blue)"power rule}$

•color(white)(x)int(ax^n)dx=a/(n+1)x^(n+1);n!=-1

$= \frac{4}{6} {x}^{6} - \frac{7}{5} {x}^{5} + \frac{3}{4} {x}^{4} - \frac{8}{3} {x}^{3} - \frac{4}{2} {x}^{2} + x + c$

$= \frac{2}{3} {x}^{6} - \frac{7}{5} {x}^{5} + \frac{3}{4} {x}^{4} - \frac{8}{3} {x}^{3} - 2 {x}^{2} + x + c$

$\text{where c is the constant of integration}$