Evaluate the integral. HINT [See Example 3.] (Remember to use ln |u| where appropriate.)
(4/x - 1/4x^5 + 1/2x^7)dx
I read that as
(4/x - (1/4)x^5 + (1/2)x^7) dx
then integral would be
4ln|x| - (1/24)x^6 + (1/16)x^8 + c, where c is a constant.
no it is just as the denominator is one with the variable 4x^5 and 2x^7 not 1/25 raised to the 6th or 1/2 raised to the 7th
then you must put brackets around it like this ....
(4/x - 1/(4x^5) + 1/(2x^7))dx
integral would be
4ln|x| + 1/(16x^4) - 1/(12x^6) + c
To evaluate the integral ∫ (4/x - 1/4x^5 + 1/2x^7)dx, we can split it into three separate integrals.
First, let's integrate 4/x. This can be written as 4 * x^(-1).
The integral of x^n dx (where n is any real number except -1) is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
Applying this to our current problem, we have:
∫ (4/x) dx = 4 * ∫ x^(-1) dx
Using the formula, we get:
4 * ((1/(1+(-1))) * x^(1+(-1))) + C
= 4 * (1/0) * x^0 + C
= 4 * (1/0) + C
= undefined + C (Since 1/0 is undefined)
= undefined
So, the integral for 4/x is undefined.
Moving on to the second part of the integral, let's integrate -1/(4x^5).
Using the formula, we have:
∫ (-1/(4x^5)) dx = (-1/4) * ∫ x^(-5) dx
Again, using the formula, we get:
(-1/4) * ((1/(1+(-5))) * x^(1+(-5))) + C
= (-1/4) * (1/(-4)) * x^(-4) + C
= (1/16) * x^(-4) + C
Finally, for the last part of the integral, let's integrate 1/(2x^7).
Using the formula, we have:
∫ (1/(2x^7)) dx = (1/2) * ∫ x^(-7) dx
Using the formula, we get:
(1/2) * ((1/(1+(-7))) * x^(1+(-7))) + C
= (1/2) * (1/(-6)) * x^(-6) + C
= (-1/12) * x^(-6) + C
Putting it all together, the integral becomes:
∫ (4/x - 1/4x^5 + 1/2x^7)dx = undefined + (1/16) * x^(-4) + (-1/12) * x^(-6) + C.