Integrate (x-x³)/x³ dx
We first split the fraction into two parts:
(x/x³) - (x³/x³)
The second term simplifies to:
1
For the first term, we can simplify the fraction by factoring out an x from the numerator:
x/x³ = 1/x²
So the integral becomes:
∫(1/x² - 1) dx
Integrating each term separately:
∫(1/x²) dx = -1/x
∫(-1) dx = -x
Putting it all together, the original integral is:
-1/x + (-x) + C
where C is the constant of integration.
To integrate the expression (x - x^3) / x^3, we can split it into two separate terms and integrate each term individually.
First, let's rewrite the expression as x/x^3 - x^3/x^3:
∫ (x - x^3) / x^3 dx = ∫ x/x^3 dx - ∫ x^3/x^3 dx
Now, let's simplify each term separately.
For the first term, ∫ x/x^3 dx, we can simplify it as:
= ∫ x^(1-3) dx
= ∫ x^-2 dx
To integrate x^-2, we can use the power rule for integration. Adding 1 to the exponent, we get:
= x^(-2+1) / (-2+1) + C
= x^-1 / -1 + C
= -1/x + C1
For the second term, ∫ x^3/x^3 dx, we have:
= ∫ 1 dx
= x + C2
Now, let's put everything together:
∫ (x - x^3) / x^3 dx = -1/x + x + C
Therefore, the integral of (x - x^3) / x^3 is -1/x + x + C, where C represents the constant of integration.