Question

integral log natural x over x dx is equal to ​

Answer 1

EXPLANATION.

⇒ ∫㏒(x)/x dx.

As we know that,

By using substitution method in this equation, we get.

⇒ log(x) = t.

Differentiate w.r.t x, we get.

⇒ dx/x = dt.

⇒ ∫t dt.

⇒ t²/2 + c.

Put the value of t = ㏒(x) in the equation, we get.

⇒ ㏒(x)²/2 + 2.

                                                                                                                         

MORE INFORMATION.

(1) = ∫sin x dx = - cos x + c.

(2) = ∫cos x dx = sin x + c.

(3) = ∫tan x dx = ㏒(sec x) + c = - ㏒(cos x) + c.

(4) = ∫cot x dx = ㏒(sin x) + c.

(5) = ∫sec x dx = ㏒(sec x + tan x ) + c = - ㏒(sec x - tan x) + c = ㏒tan(π/4 + x/2) + c.

(6) = ∫cosec x dx = - ㏒(cosec x + cot x) + c = ㏒(cosec x - cot x) + c = ㏒tan(x/2) + c.

(7) = ∫sec x tan x dx = sec x + c.

(8) = ∫cosec x cot x dx = - cosec x + c.

(9) = ∫sec²xdx = tan x + c.

(10) = ∫cosec²xdx = - cot x + c.