Question

integrate the following

Answer 1

$\huge \underline \mathfrak {Solution:-}$

Integration is the operation of finding the integral of a function or equation. It is especially used in solving a differential equation.

It is also known as anti derivative Because it is the opposite of derivative.

Derivative and antidervative, both are the parts of calculas.

It helps to find a function from its derivative.

Integral is of two types.

(I) Definite integral

(ii) indefinite integral

Answer 2

$\implies\displaystyle \int \dfrac{1 + \cot(x) }{x + log( \sin(x) ) } dx$

We will solve the above question of integration using substitution method :-

$let \: \: x + log( \sin(x) ) = t$

$(1 + \dfrac{1\times\cos \: x}{ \sin \: x } ) dx= dt$

cos x / sinx = cotx

$(1 + {\cot \: x} ) dx= dt$

therefore we get :-

$\implies\displaystyle \int \dfrac{dt }{t}$

$\implies log(t) + c$

where c is constant.

Now put the value of t :-

$\implies log( x + log( \sin(x) ) + c$

ANSWER :

$\implies log( x + log( \sin(x) ) + c$

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Learn more :-

∫ 1 dx = x + C

∫ sin x dx = – cos x + C

∫ cos x dx = sin x + C

∫ sec2 dx = tan x + C

∫ csc2 dx = -cot x + C

∫ sec x (tan x) dx = sec x + C

∫ csc x ( cot x) dx = – csc x + C

∫ (1/x) dx = ln |x| + C

∫ ex dx = ex+ C

∫ ax dx = (ax/ln a) + C

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