Sigma x-x³/x³ dx
We can solve this integral using u-substitution.
Let u = x^3, then du/dx = 3x^2, or dx = du/3x^2.
Replacing x and dx with u and du/3x^2, respectively, we get:
∫(x-x³)/x³ dx = ∫(1-u)/u du/3x^2
= (1/3) ∫(u^-1 - 1) du
= (1/3) ln|u| - (1/3)u + C
Substituting back x^3 for u, we get:
(1/3) ln|x^3| - (1/3)x^3 + C
Simplifying this expression, we get:
(1/3) ln|x³| - (1/3)x³ + C
and of course,
(1/3) ln|x³ = ln|x|
Yes, that's correct! You can simplify (1/3) ln|x³| to ln|x| by using the property of logarithms that states ln(a^b) = b ln(a), so ln|x³| = 3 ln|x|, and (1/3) ln|x³| simplifies to ln|x|.
To solve the integral ∫(x - x³)/x³ dx, we can break it down into two separate integrals:
1. Integral of x/x³ dx:
Using the rule ∫(1/x) dx = ln|x| + C, we can rewrite the first integral as:
∫(x/x³) dx = ∫(1/x²) dx = -1/x + C₁
2. Integral of -x³/x³ dx:
The x³ in the numerator and denominator cancel out, leaving us with:
∫(-x³/x³) dx = ∫(-1) dx = -x + C₂
Now, we can combine both integrals:
∫(x - x³)/x³ dx = ∫(x/x³) dx + ∫(-x³/x³) dx
= -1/x + C₁ - x + C₂
= -1/x - x + C
So, the solution to the integral is -1/x - x + C, where C is the constant of integration.