How do you integrate by substitution int x/(sqrt(1-x^2) dx∫x√1−x2dx?

2 Answers
Roy E.
Jan 27, 2017

You don't. By inspection, the answer is (-1/2)sqrt(1-x^2)+c(−12)√1−x2+c

Explanation:

It's of the form (f'(x))/sqrt(f(x)). In other words, the thing on the top is (nearly) the derivative of the thing under the square root. When you differentiate sqrt(f(x) you get (1/2)(f'(x))/sqrt(f(x)) (chain rule).

sente
Jan 27, 2017

intx/sqrt(1-x^2)dx=-sqrt(1-x^2)+C

Explanation:

By substitution:

Let u = 1-x^2. Then du = -2xdx. Substituting, we get

intx/sqrt(1-x^2)dx = -1/2int1/sqrt(1-x^2)(-2x)dx

=-1/2int1/sqrt(u)du

=-1/2intu^(-1/2)du

=-1/2((u^(-1/2+1))/(-1/2+1))+C

=-1/2((u^(1/2))/(1/2))+C

=-sqrt(u)+C

=-sqrt(1-x^2)+C

Related questions
See all questions in Integration by Substitution
Impact of this question
4315 views around the world
You can reuse this answer
Creative Commons License