The integral of sqrt(x)(sqrt(x)+1) dx
A. 2(x^3/2+x)+c
B. x^2/2+x+c
C. 1/2(sqrt(x)+1)^2+c
D. x^2/2+2x^3/2/3+c
E. x+2sqrt(x)+c
√x(√x+1) = x + √x
surely you can do that...
Not distribute. The integral. I'm confused on u-substitution.
Nevermind. Sorry
To find the integral of the given expression, we can use the power rule for integrals and apply it to each term separately.
First, let's simplify the expression:
∫(sqrt(x)(sqrt(x) + 1)) dx
We can distribute the sqrt(x) to both terms inside the parentheses:
∫(sqrt(x)*sqrt(x) + sqrt(x)*1) dx
This simplifies to:
∫(x + sqrt(x)) dx
Using the power rule, we can integrate each term:
∫x dx = (1/2)x^2 + C (where C is the constant of integration)
∫sqrt(x) dx = (2/3)(sqrt(x))^3/2 + C = (2/3)x^3/2 + C
Now we can put the results together:
∫(sqrt(x)(sqrt(x) + 1)) dx = (1/2)x^2 + (2/3)x^3/2 + C
Comparing this with the answer choices:
A. 2(x^3/2 + x) + C (not the correct answer)
B. x^2/2 + x + C (not the correct answer)
C. 1/2(sqrt(x) + 1)^2 + C (not the correct answer)
D. x^2/2 + 2x^3/2/3 + C (not the correct answer)
E. x + 2sqrt(x) + C (not the correct answer)
Therefore, the correct answer is option: D. x^2/2 + 2x^3/2/3 + C.