Mathematics

# $\int {\frac{{2x - 1}}{{\sqrt {9{x^2} - 4} }}dx}$

##### SOLUTION

Consider the given integral.

$I=\int{\dfrac{2x-1}{\sqrt{9{{x}^{2}}-4}}}dx$

$I=\int{\dfrac{2x}{\sqrt{9{{x}^{2}}-4}}}dx-\int{\dfrac{1}{\sqrt{9{{x}^{2}}-4}}}dx$

$I={{I}_{1}}-{{I}_{2}}$      …….. (1)

So,

${{I}_{1}}=\int{\dfrac{2x}{\sqrt{9{{x}^{2}}-4}}}dx$

Let $t=9{{x}^{2}}$

$\dfrac{dt}{dx}=9\times 2x$

$\dfrac{dt}{9}=2xdx$

Therefore,

${{I}_{1}}=\dfrac{1}{9}\int{\dfrac{1}{\sqrt{t-4}}}dt$

${{I}_{1}}=\dfrac{1}{9}\left[ 2\sqrt{t-4} \right]+C$

On putting the value of t, we get

${{I}_{1}}=\dfrac{2}{9}\left[ \sqrt{9{{x}^{2}}-4} \right]+C$

Now,

${{I}_{2}}=\int{\dfrac{1}{\sqrt{9{{x}^{2}}-4}}}dx$

${{I}_{2}}=\dfrac{1}{3}\int{\dfrac{1}{\sqrt{{{x}^{2}}-{{\left( \dfrac{2}{3} \right)}^{2}}}}}dx$

We know that

$\int{\dfrac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\log \left[ x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right]+C$

Therefore,

${{I}_{2}}=\dfrac{1}{3}\left[ \log \left( x+\sqrt{{{x}^{2}}-{{\left( \dfrac{2}{3} \right)}^{2}}} \right) \right]+C$

${{I}_{2}}=\dfrac{1}{3}\left[ \log \left( x+\sqrt{{{x}^{2}}-\dfrac{4}{9}} \right) \right]+C$

From equation (1), we get

$I=\dfrac{2}{9}\left[ \sqrt{9{{x}^{2}}-4} \right]-\dfrac{1}{3}\left[ \log \left( x+\sqrt{{{x}^{2}}-\dfrac{4}{9}} \right) \right]+C$

Hence, the value is $\dfrac{2}{9}\left[ \sqrt{9{{x}^{2}}-4} \right]-\dfrac{1}{3}\left[ \log \left( x+\sqrt{{{x}^{2}}-\dfrac{4}{9}} \right) \right]+C$.

Its FREE, you're just one step away

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

#### Realted Questions

Q1 Subjective Hard
Prove that $\displaystyle \int \sqrt{x^2 - a^2} dx = \dfrac{x}{2} \sqrt{x^2 - a^2} - \dfrac{a^2}{2} \log |x + \sqrt{x^2 - a^2}| + c$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int (e^{a\ln{x}}+e^{x\ln{a}})dx$, where $x>0,\ a>0$
• A. $\dfrac{x^{a+1}}{a+1}+a^{x}\ln{a}+c$
• B. $\dfrac{x^{a+1}}{a+1}+\dfrac{a^{x}}{\ln{a}}+c$
• C. None of the above
• D. $x^{a+1}+\dfrac{a^{x}}{\ln{a}}+c$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle \int \frac{x^{4} - 4}{x^{2} \sqrt{4 + x^{2} + x^{4}}} dx$
• A. $\sqrt{4 + x^{2} + x^{4}} + c$
• B. $\displaystyle \frac{\sqrt{4 + x^{2} + x^{4}}}{2} + c$
• C. $\displaystyle \frac{\sqrt{4 + x^{2} + x^{4}}}{2\, x} + c$
• D. $\displaystyle \frac{\sqrt{4 + x^{2} + x^{4}}}{x} + c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral
$\int { \cfrac { 1+\cot { x } }{ x+\log { \sin { x } } } } dx\quad$

$\displaystyle\int \sqrt{1+\sin x}f(x)dx=\dfrac{2}{3}(1+\sin x)^{3/2}+c$, then $f(x)$ equals?