Mathematics

# Solve :$\displaystyle \int { \cfrac { x-1 }{ \sqrt { x+4 } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{x-1}{\sqrt{x+4}}dx}$

$=\displaystyle\int{\dfrac{x+4-5}{\sqrt{x+4}}dx}$

$=\displaystyle\int{\dfrac{x+4}{\sqrt{x+4}}dx}-5\displaystyle\int{\dfrac{dx}{\sqrt{x+4}}}$

$=\displaystyle\int{{\left(x+4\right)}^{1-\frac{1}{2}}}-5\displaystyle\int{{\left(x+4\right)}^{\frac{-1}{2}}dx}$

$=\displaystyle\int{{\left(x+4\right)}^{\frac{1}{2}}}-5\displaystyle\int{{\left(x+4\right)}^{\frac{-1}{2}}dx}$

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

$=\dfrac{{\left(x+4\right)}^{\frac{1}{2}+1}}{\dfrac{1}{2}+1}-5\dfrac{{\left(x+4\right)}^{\frac{-1}{2}+1}}{\dfrac{-1}{2}+1}+c$

$=\dfrac{{\left(x+4\right)}^{\frac{3}{2}}}{\dfrac{3}{2}}-5\dfrac{{\left(x+4\right)}^{\frac{1}{2}}}{\dfrac{1}{2}}+c$

$=\dfrac{2}{3}{\left(x+4\right)}^{\frac{3}{2}}-10{\left(x+4\right)}^{\frac{1}{2}}+c$

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

#### Realted Questions

Q1 Single Correct Hard
Solve $\displaystyle \int \dfrac{1}{2x^2 - 3} dx =$
• A. $\dfrac{1}{\sqrt 6} log \vert{\dfrac{\sqrt 2 x - \sqrt 3}{\sqrt 2 x + \sqrt 3}}\vert + c$
• B. $\dfrac{1}{2 \sqrt 6} log \vert{\dfrac{\sqrt 2 x + \sqrt 3}{\sqrt 2 x - \sqrt 3}}\vert + c$
• C. $\dfrac{1}{\sqrt 6} log \vert{\dfrac{\sqrt 2 x + \sqrt 3}{\sqrt 2 x - \sqrt 3}}\vert + c$
• D. $\dfrac{1}{2 \sqrt 6} log \vert{\dfrac{\sqrt 2 x - \sqrt 3}{\sqrt 2 x + \sqrt 3}}\vert + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve
$I=\displaystyle \int (\sin 3x+ \cos 4\ x\ )dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate $\displaystyle\int _{ 0 }^{ 1 }{ \sin ^{ -1 }{ \left( \dfrac { 2x }{ 1+{ x }^{ 2 } } \right) dx } }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Evaluate: $\displaystyle\int \sqrt { \dfrac { 1 - x } { 1 + x } } d x$
• A. $\cos^{-1}x+\sqrt{1-x^2}+c$
• B. $\sin^{-1}x-\sqrt{1-x^2}+c$
• C. None of these
• D. $\sin^{-1}x+\sqrt{1-x^2}+c$

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