Mathematics

Solve :$\displaystyle\int { \cfrac { x }{ \sqrt { x+4 } } } dx$

SOLUTION
$\displaystyle\int{\dfrac{x}{\sqrt{x+4}}dx}$

$=\displaystyle\int{\dfrac{x+4-4}{\sqrt{x+4}}dx}$

$=\displaystyle\int{\dfrac{x+4}{\sqrt{x+4}}dx}-4\displaystyle\int{\dfrac{dx}{\sqrt{x+4}}}$

$=\displaystyle\int{{\left(x+4\right)}^{1-\frac{1}{2}}dx}-4\displaystyle\int{{\left(x+4\right)}^{-\frac{1}{2}}dx}$

$=\displaystyle\int{{\left(x+4\right)}^{\frac{1}{2}}dx}-4\displaystyle\int{{\left(x+4\right)}^{-\frac{1}{2}}dx}$

We know that $\displaystyle\int{{\left(ax+b\right)}^{n}}=\dfrac{1}{a\left(n+1\right)}{\left(ax+b\right)}^{n+1}$

$=\dfrac{{\left(x+4\right)}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}-4\dfrac{{\left(x+4\right)}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+c$

$=\dfrac{{\left(x+4\right)}^{\frac{3}{2}}}{\dfrac{3}{2}}-4\dfrac{{\left(x+4\right)}^{\frac{1}{2}}}{\dfrac{1}{2}}+c$

$=\dfrac{2}{3}{\left(x+4\right)}^{\frac{3}{2}}-8{\left(x+4\right)}^{\frac{1}{2}}+c$

$=\dfrac{2}{3}\left(x+4\right){\left(x+4\right)}^{\frac{1}{2}}-\dfrac{24}{3}{\left(x+4\right)}^{\frac{1}{2}}+c$

$=\dfrac{2}{3}{\left(x+4\right)}^{\frac{1}{2}}\left(x+4-12\right)+c$

$=\dfrac{2}{3}{\left(x+4\right)}^{\frac{1}{2}}\left(x-8\right)+c$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

Realted Questions

Q1 Single Correct Medium
$\displaystyle \int^{\pi}_0 \frac{xdx}{a^2\cos^2x+b^2\,\sin^2x}$
• A. $\dfrac{\pi}{2ab}$
• B. $\dfrac{\pi}{ab}$
• C. $\dfrac{\pi^2}{2ab}$
• D. $\dfrac{\pi^2}{ab}$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int \frac{1}{x(x^n+1)}dx$.

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
If $I=\int \sec^{2} x \mathrm{cosec}^{4}xdx = A\cot^{3}x+B\tan x + C \cot x+D$ then
• A. $B=2$
• B. none of these
• C. $A=-\dfrac{1}{3}$
• D. $C=-2$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate $\displaystyle\int e^x\left ( \frac{1 + \sin\, x}{1 + \cos\, x} \right ) dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Subjective Medium
Solve $\displaystyle\int \dfrac {x^{2}}{(x^{2}+1)(x^{2}+4)}dx$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020