evaluate integral
definite integral sign a=1 b=8
4(x^(2/3)+14)^3/((x^(1/3))) dx
I get u= x^2/3 + 14
then du= 2/3 * x^(-1/3) dx
6 { u^4/4 | a=1 b=8
Then I get 3u^4/2 | a=1 b=8
2048-1.5= 2046.5
Is this correct. thank you all.
I did it wrong. The u= x^2/3 + 14
so my new calculation becomes 98304-75937.5= 22366.5
is that right? thanks
Your substitution for u is correct, so you wind up with
∫6u^3 du
Since u=x^2/3 + 14,
x=1 ==> u=15
x=8 ==> u=18
and you have
3u^4/2 [15,18]
= 157464 - 75937.5 = 81526.5
Not sure where you got 98304
I see what i did wrong. thanks!
To evaluate the definite integral of the given function, let's go step by step.
1. First, let's substitute u = x^(2/3) + 14. This substitution simplifies the integral.
Now, we need to find the derivative of u with respect to x to find du in terms of dx.
Taking the derivative of u = x^(2/3) + 14 with respect to x, we get:
du/dx = (2/3)x^(-1/3)
2. Next, we'll substitute du = (2/3)x^(-1/3) dx into the integral.
So, the integral becomes:
∫ 4(u^3)/(u^(2/3)) du
Cancel out u^(2/3) from the numerator and denominator:
∫ 4u^(3 - 2/3) du
∫ 4u^(7/3) du
3. To integrate, add 1 to the exponent and divide by the new exponent:
(4/(7/3 + 1)) * u^(7/3 + 1)
Simplify further:
(4/10/3) * u^(10/3)
(12/10) * u^(10/3)
(6/5) * u^(10/3)
4. Now, substitute back u = x^(2/3) + 14:
(6/5) * (x^(2/3) + 14)^(10/3)
5. Evaluate the definite integral by plugging in the limits:
∫[a=1, b=8] (6/5) * (x^(2/3) + 14)^(10/3) dx
Plug in 8:
[(6/5) * (8^(2/3) + 14)^(10/3)] - Plug in 1:
[(6/5) * (1^(2/3) + 14)^(10/3)]
Simplifying:
[(6/5) * (8^(2/3) + 14)^(10/3)] - (6/5) * (15^(10/3))
Now you can use a calculator to evaluate the expression and find the result.