A "done by parts" integral is a method used to evaluate integrals that cannot be solved by other techniques, such as substitution or the power rule. It involves using the product rule of derivatives to break down the integral into smaller, more manageable parts.
In the "done by parts" method, the part that is integrated is typically the part that becomes simpler after each integration, while the part that is differentiated is typically the part that becomes more complicated after each differentiation. This allows for the integral to eventually be solved.
The "done by parts" method is best used when the integral involves a product of two functions, and neither substitution nor the power rule can be applied. It can also be used when the integral involves a function raised to a power, such as x2 or sin(x).
Sure, an example would be solving ∫x*sin(x)dx. Using the "done by parts" method, we would let u = x and dv = sin(x)dx. This means du = dx and v = -cos(x). Plugging these into the integral formula, we get ∫x*sin(x)dx = -x*cos(x) + ∫cos(x)dx. This integral can now be solved using substitution or the power rule.
Yes, the general formula for "done by parts" integrals is ∫u*dv = uv - ∫v*du. This formula can be used to evaluate any integral that can be solved using the "done by parts" method.