\intx^3\sqrt(1-x^2)dx?
I tried to get the answer found on Symbolab, but instead I got:
-1/3(1-x^2)^(3/2)+1/5(1-x^2)^(5/2)+C
computer
I tried to get the answer found on Symbolab, but instead I got:
computer
1 Answer
Your answer is excellent and correct!
Explanation:
You're going to have to use trig substitution for this question.
Let
I = int sin^3theta sqrt(1 - sin^2theta) costheta d theta
I = int sin^3theta sqrt(cos^2theta)costheta d theta
I = int sin^3theta cos^2thetad theta
I = int sintheta(1 -cos^2theta)cos^2thetad theta
I = int (sin theta - sinthetacos^2theta)cos^2thetad theta
I = int sinthetacos^2theta - sinthetacos^4thetad theta
I= int sin thetacos^2theta d theta - int sinthetacos^4theta d theta
Let
I = -int u^2 du +int u^4 du
I = -1/3u^3 + 1/5u^5 + C
I = -1/3cos^3theta + 1/5cos^5theta + C
Recall from our initial substitution that
I =1/5(1 - x^2)^(5/2) -1/3(1- x^2)^(3/2) + C
I checked and our answer is the same as the one shown on symbolab, except ours is simplified a little further.
Hopefully this helps!