Integration using parametric derivatives
In calculus, integration by parametric derivatives, also called parametric integration, is a method of Using known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution. Statement of the theorem. By using The Leibniz integral rule with the upper and lower bounds fixed we get that formula_1 It is also true for non-finite bounds. Examples. Example One: Exponential Integral. For example, suppose we want to find the integral Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is  certainly one way to evaluate it. However, we may also evaluate this by starting with a simpler integral and an added parameter, which in this case is "t" = 3: This converges only for "t" > 0, which is true of the desired integral. Now that we know we can differentiate both sides twice with respect to "t" (not "x") in order to add the factor of "x" in the original integral. This is the same form as the desired integral, where "t" = 3. Substituting that into the above equation gives the value: Example Two: Gaussian Integral. Starting with  the integral formula_7 And Taking the derivative with respect to "t" of both sides formula_8 In general, Taking the "n"-th derivative with respect to "t" gives us formula_9 Example Three: A Polynomial. Using the classical formula_10 and taking the derivative with respect to "t" We get formula_11 Example Four: Sums. The method can also b used on sums. Using the Weierstrass factorization of the sinh function formula_12 Taking a logarithm formula_13 formula_14 Letting formula_15 formula_16 External links. WikiBooks: Parametric_Integration