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    Integration by Parts


    The mathematicsmethod of integration by parts is very useful method in the integral of a product of two or more functions. This method is very helpful in comparing with another method of integration. Let we have two functions like f (x) and g (x) in product form then we have the formula for integration by parts: - Integration of f (x) * g (x) = f (x) * integration of g (x) – integration of [f’(x) * integration of g(x)]. In other words the integral of the product of two functions = first functions * integration of second function – integral of (differentiation of first function * integral of second function). Here is the most important part to choose first and second function.

    If we have two functions in product form if one function of these two functions is not directly integrable (like sin^-1 x, cos^-1 x, tan^-1 x, cot^-1 x, sec^-1 x, csc^-1 x, log x etc) these functions we take as a first part. If in product of two functions both functions are easily integrable, then we choose first functions as the derivative of this function is a simple function. Thus we can easily integrate the given function. Always product of two functions, in order to choose the first function, first we consider the inverse trigonometric function, logarithmic function, algebraic function, trigonometric function, exponential functions. This is the order to choose the first function.

    Let us see some examples of integration by parts. Example: - Integrate the given function by using integration by parts x^3 cos x. Solution: - given function in product form ( x^2 + x + 1) cos x. Here we choose first function algebraic function f (x) = x^2 + x + 1 and letsthe second function is g(x) cos x. Now using formula integration by parts: - the integral of the product of two functions = first functions * integration of second function – integral of (differentiation of first function * integral of second function).

    Integral of [(x^2 + x + 1) cos x] = (x^2 + x + 1) sin x – integral of [(2x + 1)sinx], again using integration by parts. And get integral of [(2x + 1)sinx] = - (2x + 1) cos x – integral of [(2) (-cos x) = -(2x + 1) cos x + 2 sin x, Now we plug the value of the integral of [(2x + 1)sinx] and get Integral of [(x^2 + x + 1) cos x] = (x^2 + x + 1) sin x – [-(2x + 1) cos x + 2 sin x], è Integral of [(x^2 + x + 1) cos x] = (x^2 + x + 1) sin x + (2x + 1) cos x - 2 sin x]. In this type we can integrate functions. This is how we can find the integral of many functions using integration by parts method. This method helps in finding integral of those functions which cannot be solved using the normal method.

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