How do you integrate $\int - 9 x^{2} {(- 3 x^{3} + 1)}^{3}$ $int-9x^2(-3x^3+1)^3$ using substitution?

Question

How do you integrate $\int - 9 x^{2} {(- 3 x^{3} + 1)}^{3} d x$ $int-9x^2(-3x^3+1)^3dx$ using substitution?

1588 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-07T17:58:44

$\frac{{(- 3 x^{3} + 1)}^{4}}{4} + C .$ $(-3x^3+1)^4/4+C.$

Let us write $I = \int - 9 x^{2} {(- 3 x^{3} + 1)}^{3} d x .$ $I=int-9x^2(-3x^3+1)^3dx.$

Sbstn. $(- 3 x^{3} + 1) = y \Rightarrow - 9 x^{2} d x = d y .$ $(-3x^3+1)=y rArr -9x^2dx=dy.$ Hence,

$I = \int y^{3} d y$ $I=inty^3dy$

$= \frac{y^{4}}{4}$ $=y^4/4$

$= \frac{{(- 3 x^{3} + 1)}^{4}}{4} + C .$ $=(-3x^3+1)^4/4+C.$