Find the indefinite integral and check the result by differentiation:
(integral) x^2 +2x -3
---------- dx
x^4
If you rearrange the expression from
(x²+2x-3)/x⁴
to
(x-2+2x-3-3xx-4
You can integrate term by term using standard integrals.
x-2+2x-3-3x-4
To find the indefinite integral of the function, we can use the power rule for integration, which states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1.
In this case, we have:
∫ (x^2 + 2x - 3)/(x^4) dx
Let's break this up into three separate integrals:
∫ (x^2)/(x^4) dx + ∫ (2x)/(x^4) dx - ∫ (3)/(x^4) dx
Now, let's simplify each integral and find the antiderivative for each term:
1) ∫ (x^2)/(x^4) dx:
Here, we can simplify the expression by subtracting the exponents:
∫ x^(-2) dx
Using the power rule, we add 1 to the exponent and divide by the new exponent:
= x^(-2 + 1)/(-2 + 1)
= x^(-1)/(-1)
= -1/x
2) ∫ (2x)/(x^4) dx:
We can simplify the expression by subtracting the exponents:
∫ 2x^(-3) dx
Using the power rule, we add 1 to the exponent and divide by the new exponent:
= (2/(-3 + 1)) * x^(-3 + 1)
= (2/(-2)) * x^(-2)
= -2/x^2
3) ∫ (3)/(x^4) dx:
We can rewrite this expression as 3 * (x^(-4)) and use the power rule:
∫ 3x^(-4) dx
Using the power rule, we add 1 to the exponent and divide by the new exponent:
= (3/(-4 + 1)) * x^(-4 + 1)
= (3/(-3)) * x^(-3)
= -3/x^3
Now, we can combine the three antiderivatives obtained:
∫ (x^2 + 2x - 3)/(x^4) dx = (-1/x) + (-2/x^2) + (-3/x^3)
To check the result by differentiation, we can take the derivative of the obtained result and see if it matches the original function.
Let's find the derivative of the expression:
d/dx [(-1/x) + (-2/x^2) + (-3/x^3)]
For each term, we can use the power rule in reverse to differentiate it:
1) d/dx (-1/x):
Using the power rule, we subtract 1 from the exponent and multiply by the coefficient:
= (-1) * (-1/x^2)
= 1/x^2
2) d/dx (-2/x^2):
Using the power rule, we subtract 1 from the exponent and multiply by the coefficient:
= (-2) * (-2/x^3)
= 4/x^3
3) d/dx (-3/x^3):
Using the power rule, we subtract 1 from the exponent and multiply by the coefficient:
= (-3) * (-3/x^4)
= 9/x^4
Now, let's sum up the derivatives:
= 1/x^2 + 4/x^3 + 9/x^4
If we simplify this expression, we can see that it matches the original function x^2 + 2x - 3 divided by x^4. Therefore, our result is correct.