Mathematics

# $\int {\dfrac{1}{{9{x^2} - 25}}dx = \_\_\_\_\_\_ + c.}$

$\dfrac{1}{{30}}\log \left| {\dfrac{{3x - 5}}{{3x + 5}}} \right|$

##### SOLUTION
$I=\int { \dfrac { 1 }{ {9{ x^{ 2 } }-25 } } dx }$

$I=\dfrac { 1 }{ 9 } \int { \dfrac { 1 }{ { \left( { { x^{ 2 } }-\dfrac { { 25 } }{ 9 } } \right) } } dx }$
$I=\dfrac { 1 }{ 9 } \int { \dfrac { 1 }{ { { { x^{ 2 } }-\left(\dfrac { 5 }{ 3 }\right)^2 } } } dx }$

We know that
$\int \dfrac { dx }{ x^ 2-a^2}=\dfrac{1}{2a}\log\left(\dfrac{x-a}{x+a}\right)+C$

Therefore,
$I =\dfrac { 1 }{ 9 } .\dfrac { 1 }{ { 2\times \dfrac { 5 }{ 3 } } } \log \left| { \dfrac { { x-\dfrac { 5 }{ 3 } } }{ { x+\dfrac { 5 }{ 3 } } } } \right| +C$
$I=\dfrac { 3 }{ { 90 } } \log \left| { \dfrac { { \dfrac { { 3x-5 } }{ 3 } } }{ { \dfrac { { 3x+5 } }{ 3 } } } } \right| +C$
$I =\dfrac { 1 }{ { 30 } } \log \left| { \dfrac { { 3x-5 } }{ { 3x+5 } } } \right| +C$

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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 the integrals : $\displaystyle \int \dfrac{(x^2+1)}{(x^4+1) }dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
Value of $\displaystyle \int_{0}^{\pi /4}\left ( \sqrt{\tan x}-\sqrt{\cot x} \right )\: dx$ is
• A. $\sqrt{2}\log \left ( \sqrt{2}+1 \right )$
• B. $\log \left ( \sqrt{2}+1 \right )$
• C. $\log \left ( \sqrt{2}-1 \right )$
• D. $\sqrt{2}\log \left ( \sqrt{2}-1 \right )$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
if $\int f(x) dx = f(x)$, then $\int {\left(\dfrac{f(x)}{f'(x)}\right)}$. dx is equal to
• A. log f(x) + c
• B. log F(x) + c
• C. $e^{f(x)} + c$
• D. x + c

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Prove that $\displaystyle\int^{\pi/2}_0\dfrac{\sin^{3/2}x}{(\sin^{3/2}x+\cos^{3/2}x)}dx=\dfrac{\pi}{4}$.

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$