Notes On Integration by Parts - CBSE Class 12 Maths
We try to find the integral of the product of two functions, we can do it in two ways. ∫ f(x).g(x) dx If one function is the derivative of the other, then we can use the method of substitution. If that is not the case, then we follow the method of integration by parts. Suppose u and v are two differentiable functions of a single variable x Consider ∫ uv dx d.dx (uv) = u. dv/dx = v. du/dx ∫ d(uv)/dx .dx = ∫ u. dv/dx .dx + ∫ v. du/dx . dx uv = ∫ u. dv/dx .dx + ∫ v . du/dx . dx ∫ u . dv/dx . dx = uv - ∫ v . du/dx dx u = f(x) ⇒ du/dx = f '(x) dv/dx = g(x) ⇒ v = ∫ g(x) dx ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [∫ g(x) dx] f '(x) dx ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx The integral of the product of two functions = (First function) × (Integral of the second function) - Integral of [(Differential coefficient of the first function) × (Integral of the second function)] In a given integrand, we often have to judge which function has to be taken as the first function and which one as the second. This can be easily done with the help of a precedence rule, called ILATE. Each letter in this acronym stands for a certain kind of function. 'I' stands for inverse functions. Ex: sin-1 x,cos-1 x,.... L for logarithmic functions. EX; log x,.... A for algebraic functions. Ex: ax2+bx+c, x3+3x2+2..... T for trigonometric functions. EX: sin x , cos x,.... E for exponential functions. EX : ax,ex,.... Ex: ∫ x sin x dx In this integrand, we have an algebraic function, x and a trigonometric function, sin x. According to the ILATE rule, x is considered the first function and sin x is considered the second function. ∫ log x . x3 dx So, we consider the logarithmic function as the first function and the algebraic function as the second function.

#### Summary

We try to find the integral of the product of two functions, we can do it in two ways. ∫ f(x).g(x) dx If one function is the derivative of the other, then we can use the method of substitution. If that is not the case, then we follow the method of integration by parts. Suppose u and v are two differentiable functions of a single variable x Consider ∫ uv dx d.dx (uv) = u. dv/dx = v. du/dx ∫ d(uv)/dx .dx = ∫ u. dv/dx .dx + ∫ v. du/dx . dx uv = ∫ u. dv/dx .dx + ∫ v . du/dx . dx ∫ u . dv/dx . dx = uv - ∫ v . du/dx dx u = f(x) ⇒ du/dx = f '(x) dv/dx = g(x) ⇒ v = ∫ g(x) dx ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [∫ g(x) dx] f '(x) dx ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [ f '(x) ∫ g(x) dx]  dx The integral of the product of two functions = (First function) × (Integral of the second function) - Integral of [(Differential coefficient of the first function) × (Integral of the second function)] In a given integrand, we often have to judge which function has to be taken as the first function and which one as the second. This can be easily done with the help of a precedence rule, called ILATE. Each letter in this acronym stands for a certain kind of function. 'I' stands for inverse functions. Ex: sin-1 x,cos-1 x,.... L for logarithmic functions. EX; log x,.... A for algebraic functions. Ex: ax2+bx+c, x3+3x2+2..... T for trigonometric functions. EX: sin x , cos x,.... E for exponential functions. EX : ax,ex,.... Ex: ∫ x sin x dx In this integrand, we have an algebraic function, x and a trigonometric function, sin x. According to the ILATE rule, x is considered the first function and sin x is considered the second function. ∫ log x . x3 dx So, we consider the logarithmic function as the first function and the algebraic function as the second function.

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