Integration by parts formula
Integration by parts is a method used in calculus to find the integral of a product of two functions
Integration by parts is a method used in calculus to find the integral of a product of two functions. The formula for integration by parts is given by:
∫u dv = uv – ∫v du
where u and v are functions of x, and du/dx and dv/dx are their respective derivatives.
To understand and apply this formula, follow these steps:
1. Choose u: Select one function to be u, and differentiate it to find du/dx.
2. Choose dv: Select the other function to be dv, and integrate it to find v.
3. Compute du: Differentiate u to find du/dx.
4. Compute v: Integrate dv to find v.
5. Apply the formula: Plug in the values of u, v, du/dx, and dv/dx into the integration by parts formula.
Let’s work through an example to illustrate the process:
Example: Find the integral of x sin(x) with respect to x.
Solution:
Step 1: Choose u: Let u = x, then du/dx = 1.
Step 2: Choose dv: Let dv = sin(x), then integrate to find v. Integrating sin(x) gives -cos(x), so v = -cos(x).
Step 3: Compute du: Differentiate u to find du/dx = 1.
Step 4: Compute v: Integrate dv to find v = -cos(x).
Step 5: Apply the formula: Plug in the values into the integration by parts formula.
∫x sin(x) dx = uv – ∫v du
= x(-cos(x)) – ∫(-cos(x))dx
= -x cos(x) + ∫cos(x) dx
The integral of cos(x) is sin(x), so we have:
∫x sin(x) dx = -x cos(x) + sin(x) + C
where C is the constant of integration.
That is the solution to the integral of x sin(x) with respect to x using the integration by parts formula.
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