Mathematics

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

##### SOLUTION
$\displaystyle \int \frac{x^2+x+5}{3x+2}dx$
$\displaystyle \Rightarrow \int \frac{x^{2}}{3x+2}dx+\int \frac{x}{3x+2}dx+ \int \frac{5}{3x+2}dx$
$\displaystyle \Rightarrow \int (\frac{x}{3}-\frac{2}{9}+\frac{4}{9(3x+2)})dx$
$\displaystyle I_1 = \int \frac{x}{3}dx+\int \frac{-2}{9}dx+\int \frac{4}{9(3x+2)}dx$
$\displaystyle \Rightarrow \frac{x^2}{6}-\frac{2}{9}x+\frac{4}{9}\int \frac{1}{3x+2}dx$
$\displaystyle \frac{4}{9.3}ln\left | 3x+2 \right |$
$\displaystyle I_2 : \int \dfrac{x}{3x+2}dx$
$\displaystyle \int \dfrac{1}{9}(\dfrac{U-2}{U})dv$
$\displaystyle \frac{1}{9}U-\dfrac{2}{9}ln|u|\Rightarrow \dfrac{1}{9}((3x+2)-2ln/3x+2)$
$\displaystyle I_3 \Rightarrow \int \dfrac{5}{3x+2}dx$
$\displaystyle = \dfrac{5}{3}ln |3x+2|$
$\displaystyle I = \dfrac{x^{2}}{6}-\dfrac{2x}{9}+\dfrac{4}{27}ln|3x+2|+\dfrac{1}{9}(3x+2-2n(3x+2)+\dfrac{5}{3} ln |3x+2|$
$\displaystyle \Rightarrow \frac{x^{2}}{6}+\dfrac{1}{9}x+\dfrac{43}{27}ln|3x+2|+\dfrac{2}{9}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
Evaluate :
$\int x^x (1+\log x) \ dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \overset{1}{\underset{0}{\int}} \sin \left(2\tan^{-1} \sqrt{\dfrac{1 - x}{1 + x}}\right)dx =$
• A. $\dfrac{\pi}{2}$
• B. $\dfrac{\pi}{3}$
• C. $\dfrac{\pi}{4}$
• D. $\pi$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
Let $f(x)$ be a function satisfying $f^\prime(x)=f(x)$ with $f(0)=1$ and $g$ be the function satisfying $f(x)+g(x)=x^2$. The value of the integral $\displaystyle\int_0^1{f(x)g(x)dx}$ is
• A. $e + \dfrac{{{e^2}}}{2} - \dfrac{3}{2}$
• B. $e - \dfrac{{{e^2}}}{2} - \dfrac{3}{2}$
• C. $e - \dfrac{{{e^2}}}{2} - \dfrac{5}{2}$
• D. $e + \frac{{{e^2}}}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate the following integral:
$\int { \cfrac { 2x+3 }{ \sqrt { { x }^{ 2 }+4x+5 } } } dx$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$