Mathematics

# Solve:$\displaystyle \int \dfrac{1 + x + \sqrt{x + x^2}}{\sqrt{x} + \sqrt{1 + x}}dx$ is equal to

$\dfrac{2}{3}(1 + x)^{3/2} + C$

##### SOLUTION
$\displaystyle\int \dfrac{(1+x)+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}dx$
$=\displaystyle\int \dfrac{(1+x)+\sqrt{x(1+x)}}{\sqrt{x}+\sqrt{1+x}}\times \dfrac{(\sqrt{x}-\sqrt{1+x})}{(\sqrt{x}-\sqrt{1+x})}dx$
$=\displaystyle\int \dfrac{(1+x)+\sqrt{x}\sqrt{(1+x)}}{(x-(1+x))}(\sqrt{x}-\sqrt{1+x})dx$
$=\displaystyle\int \dfrac{\sqrt{x}(1+x)-\sqrt{1+x}(1+x)+x\sqrt{1+x}-\sqrt{x}(1+x)}{(-1)}dx$
$=\displaystyle\int \dfrac{\sqrt{1+x}(x-1-x)}{(-1)}dx$
$=\displaystyle\int \sqrt{1+x}dx$
$=\dfrac{2(x+1)^{\dfrac{3}{2}}}{3}+C$
$\therefore \displaystyle\int\dfrac{(1+x)+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}}dx=\dfrac{2(x+1)^{\dfrac{3}{2}}}{3}+C$.

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

#### Realted Questions

Q1 Subjective Medium
Evaluate $\displaystyle \int {\dfrac {y(1+y)}{1-y}}dy$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac{d\left ( x^{2}+1 \right )}{\sqrt{x^{2}+2}}$ is equal to
• A. $\displaystyle \sqrt{x^{2}+2}+k$
• B. $\displaystyle \left(\frac{1}{x^{2}+2}\right) ^{\tfrac 32}+k$
• C. none of these
• D. $\displaystyle 2\sqrt{x^{2}+2}+k$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Integrate:$\displaystyle\int{{\tan}^{3}{x}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Evaluate : $\displaystyle\int^2_0\sqrt{6x+4}dx$
• A. $\dfrac{64}{9}$
• B. $7$
• C. $\dfrac{60}{9}$
• D. $\dfrac{56}{9}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
If $\dfrac{d}{dx}f(x)=g(x)$ then $\displaystyle \int_{a}^{b}f(x)g(x)dx=$
• A. $\dfrac{f(b)-f(a)}{2}$
• B. $\dfrac{f(a)-f(b)}{2}$
• C. $\dfrac{{f}^{2}(a)-{f}^{2}(b)}{2}$
• D. $\dfrac{{f}^{2}(b)-{f}^{2}(a)}{2}$