Integration by Parts
uv – int(vdu)
Integration by Parts is a method used to integrate a product of two functions. If we have two functions u and v that are differentiable, the formula for Integration by Parts is:
∫u dv = uv – ∫v du
The Integration by Parts formula essentially involves breaking down the integrand into two parts and integrating one of the parts while differentiating the other. This technique is useful when we have a product of functions and want to find the antiderivative of the product.
Here are the steps to follow when using Integration by Parts:
Step 1: Choose u and dv. Decide which function to integrate and which function to differentiate. It is also important to choose dv such that its antiderivative is easy to find.
Step 2: Differentiate u to find du. This is done using standard differentiation rules.
Step 3: Integrate dv to find v. This is done by finding the antiderivative of dv, which is often easier than finding the antiderivative of u.
Step 4: Apply the Integration by Parts formula to find the antiderivative of the product of the two functions. The formula is:
∫u dv = uv – ∫v du
Step 5: Simplify and, if possible, evaluate the integral.
It is worth noting that with repeated application of Integration by Parts, we can sometimes simplify an integration problem and solve it.
Overall, Integration by Parts is a useful technique when we are faced with a product of two functions and helps us to integrate such terms with ease.
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