How can I calculate $int_-1^0(x^2*sqrt(1+x^3))dx$ ?

Question

How can I calculate $int_-1^0(x^2*sqrt(1+x^3))dx$

1155 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-13T23:06:37

$int_(-1)^0 x^2sqrt(1+x^3) = 2/9$

Note that:

$d/dx (1+x^3) = 3x^2$

so:

$int_(-1)^0 x^2sqrt(1+x^3) = 1/3 int_(-1)^0 sqrt(1+x^3) d(1+x^3)$

$int_(-1)^0 x^2sqrt(1+x^3) = 1/3 [(1+x^3)^(3/2)/(3/2)]_(-1)^0$

$int_(-1)^0 x^2sqrt(1+x^3) = 2/9 [(1+0)^(3/2)-(1+(-1)^3)^(3/2)] =2/9$