Find the antiderivative by hand in each case.
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?
a. correct
b.Somehow I get the negative of your result. See http://calc101.com/webMathematica/derivatives.jsp#topdoit
For problem (A):
1. Start with the given integral: ∫ x * sqrt(10 + x^2) dx.
2. Let u = 10 + x^2. This choice allows us to simplify the integrand.
3. Differentiate u with respect to x to find du/dx. We have du/dx = 2x.
4. Solve for dx in terms of du by dividing both sides of the equation by 2x: dx = du/(2x).
5. Substitute these values into the integral: ∫ (x * sqrt(u)) * (du/(2x)).
6. Simplify the expression to get (1/2) ∫ sqrt(u) du.
7. Integrate (1/2) ∫ sqrt(u) du: (1/2) * (2/3) * u^(3/2) + C, where C is the constant of integration.
8. Replace u with 10 + x^2 to get the final antiderivative: (1/3) * (10 + x^2)^(3/2) + C.
Therefore, the correct antiderivative is (1/3) * (10 + x^2)^(3/2).
For problem (B):
1. Start with the given integral: ∫ (x/(sqrt(2 - 3x))) dx.
2. Let u = 2 - 3x. This choice allows us to simplify the integrand.
3. Differentiate u with respect to x to find du/dx. We have du/dx = -3.
4. Solve for dx in terms of du by dividing both sides of the equation by -3: dx = du/-3.
5. Substitute these values into the integral: ∫ (-((u - 2)/3)) * (u^(-1/2)) du.
6. Simplify the expression to get (-1/3) ∫ ((-u^(1/2)/3) + ((2u^(-1/2))/3)) du.
7. Integrate (-1/3) ∫ ((-u^(1/2)/3) + ((2u^(-1/2))/3)) du: (-2/27) * (2 - 3x)^(3/2) + (4/9) * (2 - 3x)^(1/2) + C, where C is the constant of integration.
Therefore, the correct antiderivative is (-2/27) * (2 - 3x)^(3/2) + (4/9) * (2 - 3x)^(1/2) + C.
Your answers for both problems are correct.