Question

Evaluate:

\(\sf \int_{0}^{1}x \sqrt{1-\sqrt{1-x^2}}\)

Answer is 4/15. Just wanted to see how you do it.

Answer 1

the generated solution uses extra substitutions but you should be able to get the point.

Answer 2

generic u sub problem

Answer 3

lol the fun is gone.

My method:
\(\sf u=1-\sqrt{1-x^2}\\ \sqrt{1-x^2}=1-u\\1-x^2=(1-u)^2\\-2x\ dx=2(1-u)(-1)du\\x\ dx= (1-u)du\)

when x=0, u=0
x=1, u=1

so..

\(\sf \int_{0}^{1}\sqrt{u}(1-u)du=\int_{0}^{1}(u^{1/2}-u^{3/2})du = \frac{2}{3}u^{3/2}-\frac{2}{5}u^{3/2}=(2/3-2/5)= 4/15\)

☺

Answer 4

hahaha noice