Find the antiderivative by hand in each case.

Question

S stands for the integral sign

I want to make sure I am doing these correctly.

A) S x*sqrt(10 + x^2) dx

So, u= 10 + x^2 
du= 2xdx 
du/2= xdx

(1/2) S sqrt(u) du 
(1/2)*((u^(3/2))/(3/2)) 
(1/2)*(2/3)*(u^(3/2)) 
(1/3)*(u^(3/2)) 
= (1/3)*(10 + x^2)^(3/2) correct/incorrect?

B) S (x/(sqrt(2 - 3x)))dx

S (x)*(2 - 3x)^(-1/2) 
u= 2 - 3x 
du= -3dx 
du/-3=dx 
x= -((u - 2)/3)

=(-1/3) S (-((u - 2)/3))*(u^(-1/2)) 
=(-1/3) S ((-((u^(1/2)/3)-((2u^(-1/2))/3)) 
=(-1/3)*[((2/9)*(u^(3/2))-((4/3)*(u^(1/2))] 
=[(-2/27)*((2 - 3x)^(3/2)) + (4/9)*((2 - 3x)^(1/2))] correct/incorrect?

bobpursley · Answer

a. correct b.Somehow I get the negative of your result. See http://calc101.com/webMathematica/derivatives.jsp#topdoit