Here f=cos, and we have g=x2 and its derivative 2x The General Form of integration by substitution is: ∫ f(g(x)).g'(x).dx = f(t).dt, where t = g(x). It means that the given integral is of the form: Here, first, integrate the function with respect to the substituted value (f(u)), and finish the process by substituting the original function g(x). The anti-derivatives of basic functions are known to us. ∫sin (x3).3x2.dx———————–(i). This integral is good to go! To learn more about integration by substitution please download BYJU’S- The Learning App. Never fear! Integration by substitution works by recognizing the "inside" function g(x) and replacing it with a variable. To understand this concept better, let us look into the examples. Once the substitution was made the resulting integral became Z √ udu. "Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an integral, but only when it can be set up in a special way. 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This method is used to find an integral value when it is set up in a unique form. Doing so, the function simplifies and then the basic formulas of integration can be used to integrate the function. We might be able to let x = sin t, say, to make the integral easier. The first and most vital step is to be able to write our integral in this form: Note that we have g(x) and its derivative g'(x). Take for example an equation having an independent variable in x, i.e. The first and most vital step is to be able to write our integral in this form: Note that we have g (x) and its derivative g' (x) By setting u = g(x), we can rewrite the derivative as d dx(F (u)) = F ′ (u)u ′. In the equation given above the independent variable can be transformed into another variable say t. Differentiation of above equation will give-, Substituting the value of (ii) and (iii) in (i), we have, Thus the integration of the above equation will give, Again putting back the value of t from equation (ii), we get. Substituting the value of (1) in (2), we have I = etan-1x + C. This is the required integration for the given function.

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