Integration by parts formula
∫udv= uv-∫vdu
The integration by parts formula is a technique used to integrate the product of two functions. It is written as:
∫ u(x) dv/dx dx = u(x) v(x) – ∫ v(x) du/dx dx
where u(x) and v(x) are functions of x, and dv/dx and du/dx are their respective derivatives with respect to x.
To use the formula, you choose u(x) and dv/dx, and then use the formula to find the integral of the product. You can then use algebra to solve for the integral you want.
Here’s an example:
∫ x e^x dx
To use integration by parts here, we choose u(x) = x and dv/dx = e^x.
Then, du/dx = 1 and v(x) = e^x.
Using the formula, we have:
∫ x e^x dx = x e^x – ∫ e^x dx
= x e^x – e^x + C
where C is the constant of integration.
So the final answer is:
∫ x e^x dx = x e^x – e^x + C
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