Mathematics

# Solve:$\displaystyle\int {\dfrac{{4x + 6}}{{2{x^2} + 5x + 3}}dx}$

##### SOLUTION
$\displaystyle I= \int \frac {4x+6}{2x^2+5x+3} dx= \int \frac {4x+6 dx}{2x^2+3x+2x+3}= \int \frac {(4x+6)dx}{(2x+3)x+1(2x+3)}$

$\displaystyle I = \int \frac {dx(4x+6)}{(x+1)(2x+3)}=\int \frac {dx(4x+4)}{(x+1)(2x+3)}+2 \int \frac {dx}{(x+1)(2x+3)}$

$=\displaystyle \int \frac {4dx}{(2x+3)}+ 2\int \frac{dx}{(x+1)(2x+3)}$

$\displaystyle I = a |n|2x+3| = 2 \left [ \frac {2}{(2x+3)}- \frac {1}{(x+1)} \right ]$

$\displaystyle I = 4|n|2x+3| - 4|n|2x+3|+ 2|n|x+1|+ |n|c|$

$\displaystyle I = |n|c(x+1)^2|$

$\displaystyle \therefore \int \frac {4x+6}{(2x^2+5x+3)}dx = |n|c(x+1)^2|$     c= constant.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium

$\displaystyle \int_{\pi/6}^{\pi/3}\frac{\cos^{3}x}{\sin^{3}x+\cos^{3}x}dx_{=}$
• A. $\displaystyle \frac{\pi}{3}$
• B. $\displaystyle \frac{-\pi}{2}$
• C. $\displaystyle \frac{\pi}{6}$
• D. $\displaystyle \frac{\pi}{12}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int\dfrac{\sqrt{x^2+1}[\log(x^2+1)-2\log x]}{x^4}$ is equal to
• A. $\dfrac{1}{3}\left(1+\frac{1}{x^2}\right)^\cfrac32\left[ \log\left(1+\frac{1}{x^2}\right)+\frac{2}{3}\right] +C$
• B. $\dfrac{2}{3}\left(1+\frac{1}{x^2}\right)^\cfrac{1}{2}\left[ \log\left(1+\frac{1}{x^2}\right)+\frac{2}{3}\right] +C$
• C. None of these
• D. $-\frac{1}{3}\left(1+\dfrac{1}{x^2}\right)^\cfrac{3}{2}\left[ \log\left(1+\frac{1}{x^2}\right)-\frac{2}{3}\right] +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int \left ( x-2 \right )\sqrt{2x^{2}-6x+5}dx=\frac{x-3}{2}\sqrt{\left ( x-\frac{3}{2} \right )^{2}+\left ( \frac{1}{2} \right )^{2}}$ $\displaystyle +\frac{1}{2}\left ( \frac{1}{2} \right )^{2}\log \left [ \left ( x-\frac{3}{2} \right )+\sqrt{\left ( x-\frac{3}{2} \right )^{2}+\left ( \frac{1}{2} \right)^{2}}\right ].$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate: $\displaystyle \int { \cfrac { \sin { \left( \log { x } \right) } }{ x } } dx$

$\int { \cfrac { f'(x) }{ f(x) } dx } =\log { [f(x)] } +c$