ʃ dx/x^4(1/x^2-2)^2
please help i need solution..i got the answer..
answer= 1/6(1-2x^3)=c
thanks
Differentiating your answer would give us
x^2 , not anywhere near your starting integral.
You should have realized that your answer cannot be correct.
Before any tutor starts your messy expression, we cannot tell exactly what you mean
According to the way you typed it and following the order of operation I read that as
ʃ (1/x^4)(1/x^2-2)^2 dx
please confirm
(1/x^4)(1/(x^2-2))^2 dx
= (x^2 - 2)^2/x^4
= (x^4 - 4x^2 + 4)/x^4
That's just a bunch of power terms x^n.
As Reiny said, please clarify what your expression really is.
note that 1/x^2-2 can be (1/x^2) - 2 or (1/(x^2 - 2) depending on how casual you are in your use of parentheses.
To solve the integral ʃ dx/x^4(1/x^2-2)^2, we can use the substitution method. Let's start by making the substitution u = 1/x^2 - 2.
Differentiating u with respect to x gives du/dx = -2/x^3.
Now, we can solve for dx in terms of du: dx = (-x^3/2) du.
Substituting the expressions for x and dx into the integral, we have:
ʃ (-x^3/2) du / x^4 (1/x^2 - 2)^2
Next, we simplify the integral:
ʃ (-1/2) (x^-1) du / (1/x^2 - 2)^2
Recall that x^-1 is equivalent to 1/x.
ʃ (-1/2) (1/x) du / (1/x^2 - 2)^2
Now, we can simplify further:
-1/2 ʃ (1/x) du / (1/x^2 - 2)^2
Using the fact that 1/(ab) = a^-1b^-1, we can rewrite the integral as:
-1/2 ʃ (x/1) du / (1/x^2 - 2)^2
Now, we can cancel out the x and 1 from the numerator and denominator:
-1/2 ʃ du / (1/x^2 - 2)^2
The integral simplifies to:
-1/2 ʃ du / u^2
Applying the power rule of integration, the integral becomes:
-1/2 * (-1/1) u^-1 + C
Simplifying further gives:
1/2u + C
Finally, we substitute u back in terms of x:
1/2(1/x^2 - 2) + C
Which simplifies to:
(1 - 2x^3)/2x^2 + C
Multiplying everything by 1/6 to obtain the final answer, we have:
(1/6)(1 - 2x^3)/x^2 + C
Therefore, the final answer is:
(1/6)(1 - 2x^3) = C, where C is the constant of integration.