Math, asked by harshsingh2, 1 year ago

integration of {2t+3 ÷ 7t + 22} dt

Answers

Answered by MaheswariS
3

\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}

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Answered by amitnrw
4

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

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