Mathematics

# Evaluate the given integral.$\displaystyle\int{\dfrac{{\left(1+x\right)}^{2}}{\sqrt{x}}dx}$

##### SOLUTION
$I=\displaystyle\int{\dfrac{{\left(1+x\right)}^{2}}{\sqrt{x}}dx}$

$=\displaystyle\int{\dfrac{\left(1+{x}^{2}+2x\right)}{\sqrt{x}}dx}$

$=\displaystyle\int{\left({x}^{-\frac{1}{2}}+{x}^{2-\frac{1}{2}}+2{x}^{1-\frac{1}{2}}\right)dx}$

$=\displaystyle\int{\left({x}^{-\frac{1}{2}}+{x}^{\frac{3}{2}}+2{x}^{\frac{1}{2}}\right)dx}$

$=\displaystyle\int{{x}^{-\frac{1}{2}}dx}+\displaystyle\int{{x}^{\frac{3}{2}}dx}+2\displaystyle\int{{x}^{\frac{1}{2}}dx}$

$=\dfrac{{x}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+\dfrac{{x}^{\frac{3}{2}+1}}{\dfrac{3}{2}+1}+2\times\dfrac{{x}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}+c$

$=\dfrac{{x}^{\frac{1}{2}}}{\dfrac{1}{2}}+\dfrac{{x}^{\frac{5}{2}}}{\dfrac{5}{2}}+2\times\dfrac{{x}^{\frac{3}{2}}}{\dfrac{3}{2}}+c$

$=2\sqrt{x}+\dfrac{2}{5}{x}^{\frac{5}{2}}+\dfrac{4}{3}{x}^{\frac{3}{2}}+c$

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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 $I^n \, = \, {I_{0}}^{\pi/4} \, tan ^n \, X sec ^2 X dx$     then $I_1 , \, I _2 \, , \, I_3 \,$ are in
• A. G . P
• B. H . M
• C. none
• D. A. P

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int x^{n}.\log xdx=$
• A. $\displaystyle \frac{x^{n+1}}{(n+1)^{2}}(\log x-\frac{1}{(n+1)^{2}})+c$
• B. $(\displaystyle \log x-\frac{1}{(n+1)^{2}})+c$
• C. $\displaystyle \frac{x^{n+1}}{(n+1)}(\log x-\frac{1}{(n+1)^{2}})+c$
• D. $\displaystyle \frac{x^{n+1}}{(n+1)}(\log x-\frac{1}{(n+1)})+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The integral $\displaystyle \int_{\dfrac{\pi}{4}}^{\dfrac{3\pi}{4}}\dfrac{dx}{1+\cos x}$
• A. $4$
• B. $-1$
• C. $-2$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Integrate $\displaystyle \int_{2}^{3}(2x^2+1)dx$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.