Hello, I just wanted to verify this definite integral and if my answer is good:
a = 1 and b = 4
integral of
(1 - 1/x^2)(x^2 + 1/x) dx
= (x + 1/x)(x^3/3 + lnx) + C
(4 + 1/4)(64/3 + ln4) - (1 + 1)(1/3 + ln1)
= 96.56 - 0.6666 = 95.89
THank you
I do not think so. Multiply out and integrate each term.
(x^2 -1 + 1/x -1/x^3)dx
x^2 dx - dx + dx/x - dx/x^3
x^3/3 etc
integral a b dx is not integral a dx * integral b dx
To verify if your answer to the definite integral is correct, let's go through the steps to evaluate the integral and compare it with your result.
We have the integral:
∫ (1 - 1/x^2)(x^2 + 1/x) dx
Step 1: Expand the expression
= ∫ (x^2 + 1/x - x^-2 - 1/x^3) dx
Step 2: Integrate each term separately
The integral of x^2 dx is x^3/3, and the integral of 1/x dx is ln|x|.
For the other terms, we can rewrite them as x^-2 = 1/x^2 and 1/x^3 = x^-3.
Thus, the integral becomes:
∫ (x^3/3 + ln|x| - 1/x^2 - x^-3) dx
Step 3: Simplify the integral
= x^3/3 + ln|x| - 1/x + C
So, the antiderivative of the given function is (x^3/3 + ln|x| - 1/x) + C.
Next, let's evaluate the definite integral with the limits a = 1 and b = 4.
Plugging in the limits, we have:
[(4^3/3 + ln|4| - 1/4) - (1^3/3 + ln|1| - 1/1)]
Simplifying further:
[(64/3 + ln(4) - 1/4) - (1/3 + 0 - 1)]
= (64/3 + ln(4) - 1/4) - (1/3 - 1)
= 64/3 + ln(4) - 1/4 - 1/3 + 1
By computing the arithmetic, we find that the result is approximately:
= 21.333 + 1.386 - 0.25 - 0.333 + 1
= 23.719 - 0.583
= 23.136
Comparing this with your result of 95.89, there seems to be a discrepancy. Please double-check your calculations and ensure that you followed the steps correctly.
If you need any further assistance, feel free to ask!