Evaluate the following definite integral.
sqrt(11 x)(sqrt(x) + sqrt(11)) dx between(0,1)
∫√(11x)(√x + √11) dx
= ∫√11 x + 11√x dx
now it's just straight powers of x, so we have
∫[0,1] √11 x + 11√x dx
= √11 x^2/2 + 11(2/3) x^(3/2) [0,1]
= 22/3 + √11/2
To evaluate the definite integral ∫[0,1] sqrt(11x)(sqrt(x) + sqrt(11)) dx, we can use integration techniques such as substitution and the power rule.
First, let's simplify the integrand √(11x)(√x + √11). We can expand the expression inside the parentheses using the distributive property as follows:
√(11x)(√x + √11) = (√11x√x) + (√11x√11).
= 11√x + 11√11√x.
= 11√x + 11√11x.
= (11 + 11√11)√x.
Now we have the simplified integrand: (11 + 11√11)√x.
Let's evaluate the definite integral:
∫[0,1] (11 + 11√11)√x dx.
We can apply the power rule, which states that the integral of x^n is (1/(n+1))x^(n+1), to solve the integral. In this case, n = 1/2, so we have:
∫[0,1] (11 + 11√11)√x dx = (11 + 11√11) ∫[0,1] √x dx.
Using the power rule, we integrate √x:
∫[0,1] √x dx = (2/3)x^(3/2).
Now, we substitute the limits of integration 0 and 1 into the integral:
(11 + 11√11) [(2/3)(1^(3/2)) - (2/3)(0^(3/2))].
Simplifying further:
(11 + 11√11) [(2/3)(1) - (2/3)(0)].
= (11 + 11√11) (2/3)
Therefore, the value of the definite integral √(11x)(√x + √11) dx between 0 and 1 is (11 + 11√11)(2/3).