Mathematics

# Evaluate $\displaystyle\int_{0}^{2}x\sqrt{x+2}dx$

##### SOLUTION
$I=\displaystyle\int_{0}^{2}x\sqrt{x+2}dx$.

Let $x+2=t^{2}$.

Then , $dx=2t \ dt$

Also
$x=0\Rightarrow t^{2}=2\Rightarrow t=\sqrt{2}$ and, $x=2\Rightarrow t^{2}=4\Rightarrow t=2$

$\therefore I=\displaystyle\int_{\sqrt{2}}^{2}(t^{2}-2)\sqrt{t^{2}}2t dt$

$=2\displaystyle\int_{\sqrt{2}}^{2}(t^{4}-2t^{2})dt$

Using , $\displaystyle\int{{x}^{n}dx}=\dfrac{{x}^{n+1}}{n+1}+c$, we get

$=2\left[\dfrac{t^{5}}{5}-\dfrac{2t^{3}}{3}\right]_{\sqrt{2}}^{2}$

$I=2\left[\left(\dfrac{32}{5}-\dfrac{16}{3}\right)-\left(\dfrac{4\sqrt{2}}{5}-\dfrac{4\sqrt{2}}{3}\right)\right]$

$=2\left(\dfrac{16}{15}+\dfrac{8\sqrt{2}}{15}\right)$

$=\dfrac{32+16\sqrt{2}}{15}$

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

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int x^{4}\cdot \tan x^{5}dx=$
• A. $\displaystyle {\frac{1}{5}} \tan x^{5}+c$
• B. $\displaystyle {\frac{1}{5}}\log| \sec x^{5}+ \tan x^{5}|+c$
• C. $\displaystyle {\frac{1}{5}}\log| tanx^{5}|+c$
• D. $\displaystyle {\frac{1}{5}}\log|\sec x^{5}|+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Passage Medium
Consider two differentiable functions $f(x), g(x)$ satisfying $\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$ & $\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$. where $\displaystyle f(x)>0 \forall x \in R$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020

Q3 Single Correct Hard
If $f(x) = \dfrac {x + 2}{2x + 3}$, then $\displaystyle \int \left (\dfrac {f(x)}{x^{2}}\right )^{1/2} dx = \dfrac {1}{\sqrt {2}}g \left (\dfrac {1 + \sqrt {2f(x)}}{1 - \sqrt {2f(x)}}\right ) - \sqrt {\dfrac {2}{3}}h \left (\dfrac {\sqrt {3f(x)} + \sqrt {2}}{\sqrt {3f(x)} - \sqrt {2}}\right ) + c$, where
• A. $g(x) = \log |x|, h(x) = \tan^{-1}x$
• B. $g(x) = h(x) = \tan^{-1}x$
• C. $g(x) = \log|x|, h(x) = \log |x|$
• D. $g(x) = \tan^{-1} x, h(x) = \log |x|$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of $\displaystyle \lim_{n \rightarrow \infty} \sqrt{\dfrac{9(i-1)}{n}} \cdot \dfrac{9}{n} = ?$
• A. $\int_0^9 \sqrt{x}^3 dx$
• B. $\int_0^9 \sqrt{9x} dx$
• C. $\int_1^9 \sqrt{x} dx$
• D. $\int_0^9 \sqrt{x} dx$

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