Mathematics

# The value of $\int \frac{x^3}{x+2} dx$ equals

$\frac{x^3}{3} -x^2+4x - 8 log |x +2| +c$

##### SOLUTION

$\int (\frac{x^3}{x+2})dx\\Let\> x+2=t\\then \>dx =dt\\\therefore I=\int (\frac{(t-2)^3}{t})dt\\=\int (\frac{t^3-6t^2+12t-8}{t}) dt \\=\int t^2-6t+12-(\frac{8}{t})dt\\=(\frac{t^3}{3}) -(\frac{6t^2}{2}) +12t-8lot +c\\=(\frac{(x+20)^3}{3})-3(x+2)^2+12(x+2) -8log\left | x+2 \right | +c\\=(\frac{x^3}{3}) -x^2+4x-8 log\left | x+2 \right |+c$

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

#### Realted Questions

Q1 Single Correct Medium
If $\beta +2\int_0^1x^2e^{-x^2}dx=\int_0^1e^{-x^2}dx$ then the value of $\beta$ is
• A. $e$
• B. $1/2e$
• C. can not be determined
• D. $e^{-1}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\displaystyle\int \dfrac{\log x}{x} dx$ is equal to

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of integral $\displaystyle \int \frac{d\theta}{cos^3 \theta \sqrt{sin\, 2 \theta}}$ can be expressed as irrational function of $tan \theta$ as
• A. $\displaystyle \frac{\sqrt{2}}{5}\, \left ( \sqrt{tan^2\, \theta\, +\, 5}\right )\, tan \theta\, +\, c$
• B. $\displaystyle \frac{2}{5} \, ( tan^2 \theta\, +\, 5)\, \sqrt {tan \theta}\, +\, c$
• C. $\sqrt {\displaystyle \frac {2}{5}}\, (tan^2\, \theta\, +\, 5)\, \sqrt {tan\, \theta}\, +\, c$
• D. $\displaystyle \frac{\sqrt{2}}{5}\, \left ( tan^2\, \theta\, +\, 5 \right )\, \sqrt{tan \theta}\, +\, c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int^{1/\sqrt{3}}_{-1/\sqrt{3}}\frac{x^{4}}{1-x^{4}} cos^{-1} \displaystyle \frac{2x}{1+x^{2}}dx=$
• A. $\dfrac{\pi}{4}\left[\displaystyle \dfrac{1}{2}\log(2+\sqrt{3})+\dfrac{\pi}{4}-\dfrac{2}{\sqrt{2}}\right]$
• B. $\dfrac{\pi}{2}\left[\displaystyle \dfrac{1}{2}\log(2-\sqrt{3})-\dfrac{\pi}{6}+\dfrac{2}{\sqrt{3}}\right]$
• C. $\dfrac{\pi}{2}\left[\displaystyle \dfrac{1}{3}\log(3+\sqrt{3})+\dfrac{\pi}{5}-\dfrac{1}{\sqrt{3}}\right]$
• D. $\dfrac{\pi}{2}\left[\displaystyle \dfrac{1}{2}\log(2+\sqrt{3})+\dfrac{\pi}{6}-\dfrac{2}{\sqrt{3}}\right]$

$\displaystyle \int_1^2(4x^3-5x^2+6x+9)dx$