When do you use integration by parts?
The formula for integration by parts is Integral(udv/dx)dx = uv - Integral(vdu/dx)dx
You use integration by parts when you have an integral where you have to terms multiplied together ie Integral(u*dv/dx)dx eg Integral(5xex)dx.From here you need to identify what term is u and what term is dv/dx from our example integral.
A good way to do this is to use the abbreviation u = LATE, where we select our u variable in the order of what our term is. Ie u = logarithm, algebraic term, trigonometric term, exponential term. In our example, u = 5x, and we therefore select dv/dx = ex.Now, we differentiate y and integrate dv/dx in order to use the formula ie du/dx = 5 and v = ex.We now substitute in our values:5xex - Integral(5ex)dx = 5xex -5ex In this question we have indefinite limits ie we never specified them. If we had limits a and b then we would integrate this over a and b.
