So, using the formula for the integration by parts and the above mentioned formulas, we have Integral of sinx: ∫sin x dx = -cos x + C.We will also use the following formula to find the integral of x sin x: Using this sequence of preference of functions, we have f(x) = x (as x is an algebraic function) and g(x) = sin x (sin x is a trigonometric function). Here, we chose the functions f(x) and g(x) using the ILATE rule which is I - Inverse Trigonometric Function, L - Logarithmic Function, A - Algebraic Function, T - Trigonometric Function, E - Exponential Function. Now, the formula for integration by parts is given by, ∫f(x) g(x) dx = f(x) ∫g(x) dx - ∫ dx. Therefore, integration by parts is also known as the product rule of integration. We use this method to find the integral of a function that is given as the product of two functions. Now that we know that the integral of x sin x is equal to −x cos x + sin x + C, we will derive this formula using the integration by parts method of integration.
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