## Manoj Chauhan is teaching live on Unacademy Plus

INTEGRATION BY PARTS If u and v are two functions of x, then juv dx = ufvdx-Rmfvdxtdx du c dx i.e., The integral of product of two functions -(first function) x (integral of second function) integral of (differential of first function x integral of second function)

Proof For any two functions f(x) and g(x), we have {f(x), g(x)) = f(x) . { g(x)J + g(x). {f(x)) x x f(f(x).- -(g(x)) + g(x).- -(f(x)))dx = ff(x) . g(x)dx or or f(f(x). x{g(x)))dx=jf(x), g(x) dx-f(g(x).- x{f(x)))dx Let f(x) = u and dx { g(x)) = v, So that g(x) = v dx .. Juv dx = u .fvdx- vdx}.dx

Points to Consider While applying the above rule, care has to be taken in the selection of first function (u) and selection of second function (v). Normally we use the following methods: If in the product of the two functions, one of the functions is not directly integrable (eg, loglkl, sin x, cos x tam x,.. etc.) The, we take it as the first function and the remaining function is taken as the second function. eg, In the integration of Jx tan x dx, tan x is taken as the first function and x as the second function. If there is no other function, then unity is taken as the second function. eg., In the integration of 1. 2. tanx dk, tanx is taken as function and I as the second function.

If both the functions are directly integrable, then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable. Usually we use the following preference order for selecting the first function. (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent) In above stated order, the function on the left is always chosen as the first function. This rule is called as ILATE 3. ILATE dx

Example Evaluate i) x cos x dx (i) fx cos x dk (ii) x cos x dx Solution (i) Jx cos x d:x I-Ixcos xdx Applying integration by parts, 1 = x([cos x dx)-j (x) (cos x) dx) dx 1 = x sin x-| 1 . sin x dx 1=xsin x + cos x + C

(ii)I- [x? cos x dx Applying integration by parts, 1 x2( 1 cos x dx We again have to integrate J x sin x dx using integration by parts, =x2 . sin x-2jx . sin x dx = x2 sin x-21 x(.sin xdx)-1 dx dx cfsin x dx)dx | = x2 sin x-2 {-x cos x-j 1 . (-cos x) dx) 1=x2 sin x + 2x cos x _ 2 sin x + C

Example Evaluate o)I-sinx dx) Jlog kl dx Solution (i) I = | sin-i x dx = | sin-1x . I dx Here, we know by definition if integration by parts that order of preference is taken according to ILATE So, 'sin x' should be taken as first and '' as the second function to apply by parts Applying integration by parts, we get 1 dt xdx = x . sin-1 x + 1 = sin-1x . (x)-

let 1-x2- 1/2 1 = x sin-1 x + 5175+C 2 1/2

(ii) l-Ilog lxl dx = flog lxl . I dx Applying integration by parts, we get