Question

Integration of dx/3x+5 = ?
1. Not defined
2. In (3x+5) +c
3. [In (3x+5)]÷3 + c
4. 3In (3x+5) + c

Please answer with steps.
No useless answers

Answer 1

as we know integeration of 1/y is lny and integeration of 1/ny where n is any number is " lny ÷3 "
so using this identity
dx / 3x+5 =[ ln( 3x+5) ]÷ 3 + c
we divide it by 3 because 3 is the coefficient of x

so option 3 is correct

Answer 2

Answer:

$I=\ln (3x+5)\div 3+C$

C is correct

Explanation:

Given: $I=\int \dfrac{dx}{3x+5}$

Formula: $\int \dfrac{1}{x}=\ln x+C$

It is definite integral. Using substitution method to integrate it.

Let 3x+5=t

differentiate both sides w.r.t t

$3dx=dt$

$I=\int \dfrac{1}{t}\cdot \dfrac{dt}{3}$

$I=\dfrac{1}{3}\ln t+C$

where, C is constant of integration.

Put t=3x+5

$I=\dfrac{1}{3}\ln (3x+5)+C$

or

$I=\ln (3x+5)\div 3+C$

Hence, The integration of given function would be $I=\ln (3x+5)\div 3+C$