xdy%2Fdx.jpeg "(1-x^2)xdy/dx")
Integration of (1-x^2) x dy/dx
In this article, we will explore the integration of (1-x^2) x dy/dx, a fundamental concept in calculus.
What is dy/dx?
Before diving into the integration, let's take a step back and understand what dy/dx represents. dy/dx is the derivative of y with respect to x, which represents the rate of change of y with respect to x. In other words, it measures how fast y changes when x changes.
The Given Expression: (1-x^2) x dy/dx
The given expression, (1-x^2) x dy/dx, is a product of three terms: (1-x^2), x, and dy/dx. To integrate this expression, we need to use the product rule of differentiation, which states that if u and v are functions of x, then:
$\int u(x)v'(x) dx = u(x)v(x) - \int v(x)u'(x) dx$
Integration of (1-x^2) x dy/dx
Using the product rule, we can rewrite the given expression as:
$\int (1-x^2) x dy/dx = \int (1-x^2) dx - \int x dy$
Now, we can integrate each term separately:
$\int (1-x^2) dx = x - \frac{x^3}{3} + C_1$
$\int x dy = \frac{x^2y}{2} + C_2$
where C1 and C2 are constants of integration.
Final Answer
Combining the two results, we get:
$\int (1-x^2) x dy/dx = x - \frac{x^3}{3} - \frac{x^2y}{2} + C$
where C is a constant.
In conclusion, we have successfully integrated the expression (1-x^2) x dy/dx using the product rule of differentiation. The final answer is a function of x and y, with a constant term.
