Question

integration of {2t+3 ÷ 7t + 22} dt

Answer 1

$\textbf{To find:}$

$\mathsf{\displaystyle\int\,\dfrac{2t+3}{7t+22}\;dt}$

$\textbf{Solution:}$

$\textsf{Consider,}$

$\mathsf{\displaystyle\int\,\dfrac{2t+3}{7t+22}\;dt}$

$\textsf{By division, it can be written as}$

$\mathsf{=\displaystyle\int\,\dfrac{\dfrac{2}{7}(7t+22)-\dfrac{23}{7}}{7t+22}\;dt}$

$\mathsf{=\displaystyle\dfrac{2}{7}\int\;dt-\dfrac{23}{7}\int\,\dfrac{1}{7t+22}\,dt}$

$\mathsf{=\dfrac{2t}{7}-\dfrac{23}{7}{\times}\dfrac{1}{7}log|7t+22|+C}$

$\mathsf{=\dfrac{2t}{7}-\dfrac{23}{49}log|7t+22|+C}$

$\implies\mathsf{\displaystyle\int\,\dfrac{2t+3}{7t+22}\;dt=\dfrac{2t}{7}-\dfrac{23}{49}log|7t+22|+C}$

Find more:

1.Integrate of e^x sec x(1+tanx) dx

https://brainly.in/question/6040969

2.Integrate cos2x/(cosx+sinx)^2

https://brainly.in/question/833990

3.Integration to (a^x+b^x)^2/a^x b^x​

https://brainly.in/question/11131378

Answer 2

Given :   {(2t+3) ÷( 7t + 22)} dt

$\int \frac{2t+3}{7t+22} dt$

To Find : integrate

Solution:

(2t + 3)/ ( 7t + 22)

= 7(2t + 3)/7(7t + 22)

=   (14t + 21 )/ 7(7t + 22)

=  ( 14t + 44 + 21 - 44) / 7(7t + 22)

=  ( 14t + 44 -23 ) / 7(7t + 22)

=  (14t + 44)/7(7t + 22)  - 23/7(7t + 22)

=  2/7  -  23/7(7t + 22)

(2/7)∫ dt  - (23/7)∫ dt/(7t + 22)  

= 2t/7   - (23/7)   ln | 7t + 22 | / 7  + C

= 2t/7  -  23 ln | 7t + 22 | /49  + C

$\int \dfrac{2t+3}{7t+22} dt = \dfrac{2}{7} -\dfrac{23. \ln |7t+22|}{49} +C$

Learn More:

integration x^4sin (x^5+6)

brainly.in/question/23787104

integrate x³-1/x³+x dx - Brainly.in

brainly.in/question/3017912