Mathematics

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

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

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

$dx=2t dt$

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

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

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

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

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

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

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

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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
If $\displaystyle I=\int \frac{dx}{\sqrt{\left ( 1-x \right )\left ( x-2 \right )}},$ then $I$ is equal to
• A. $\displaystyle \sin^{-1}\left ( 2x+5 \right )+C$
• B. $\displaystyle \sin^{-1}\left ( 3-2x \right )+C$
• C. $\displaystyle \sin^{-1}\left ( 5-2x \right )+C$
• D. $\displaystyle \sin^{-1}\left ( 2x-3 \right )+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of $\displaystyle\int { \cfrac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } dx$ is
• A. $-\cfrac { 1 }{ { e }^{ x }+{ e }^{ -x } } +C$
• B. $\cfrac { -1 }{ { \left( { e }^{ x }+1 \right) }^{ 2 } } +C$
• C. $\cfrac { 1 }{ { e }^{ x }-{ e }^{ -x } } +C\quad \quad$
• D. $\cfrac { -{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } +C\quad$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve : $\displaystyle \int\limits_{-\pi/2}^{\pi/2} \sin^7x \ dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Solve : $\int { \dfrac { cos4\theta +1 }{ cot\theta -tan\theta } d\theta }$

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 $\displaystyle f(x)>0 \forall x \in R$