Almost! [itex](1/4)(x^2)[/itex] when x= 2 is 1. when x= 1, it is 1/4 so the last part is -(1- 1/4)= -3/4, not -5/4.ricekrispie said:
Integration by parts is a method for evaluating integrals that involve products of functions. It is based on the product rule from differentiation and involves breaking down the integral into smaller, more manageable parts.
The integration by parts method is important because it allows us to solve a wide variety of integrals that cannot be solved using other methods. It also helps us to evaluate integrals more efficiently and accurately.
You should use integration by parts when the integral involves the product of two functions and one of the functions becomes simpler after differentiating and one becomes simpler after integrating. This is known as the LIATE rule.
The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are functions and dv and du are their respective differentials. This formula is derived from the product rule for differentiation.
Sure, an example of integrating using the integration by parts method would be ∫xlnx dx. Using the LIATE rule, we let u = ln x and dv = x dx. This gives us du = 1/x and v = x^2/2. Plugging these values into the formula, we get ∫xlnx dx = x^2/2 * ln x - ∫x^2/2 * 1/x dx. Simplifying this further, we get ∫xlnx dx = x^2/2 * ln x - ∫x/2 dx. By evaluating the integrals, we get the final answer of x^2/2 * ln x - x^2/4 + C.