Mathematics

# $\displaystyle\int \frac{3x^{2}}{\sqrt{\left ( 9-16x^{6} \right )}}dx.$

$\displaystyle \frac{1}{4}\sin ^{-1}\frac{4}{3}x^{3.}$

##### SOLUTION
Let $\displaystyle I=\int \frac { 3x^{ 2 } }{ \sqrt { \left( 9-16x^{ 6 } \right) } } dx$

Put $\displaystyle 4x^{ 3 }=t\Rightarrow 12x^{ 2 }dx=dt$
Therefore
$\displaystyle I=\frac { 1 }{ 4 } \int \frac { dt }{ \sqrt { \left( 3^{ 2 }-t^{ 2 } \right) } } =\frac { 1 }{ 4 } \sin ^{ -1 } \frac { t }{ 3 } =\frac { 1 }{ 4 } \sin ^{ -1 } \frac { 4 }{ 3 } x^{ 3 }$
Hence, option 'D' is correct.

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

#### Realted Questions

Q1 Subjective Medium
Solve:
$\displaystyle\int_{0}^{3}|3x-1|\ dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Prove   $\displaystyle \int_0^{\frac {\pi}{2}}\sin^3xdx=\frac {2}{3}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle I = \int_{0}^{\pi/2} \frac{dx}{5+3\sin x} =\lambda \tan^{-1} \left(\frac{1}{2}\right )$ then
value of $\lambda$ is
• A. $1$
• B. $\displaystyle \frac{1}{3}$
• C. $\displaystyle \frac{1}{4}$
• D. $\displaystyle \frac{1}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
Which of the following functions does not appear in the primitive of $\dfrac {dx}{x+\sqrt {{x}^{2}-x+1}}$ if $t$ is a function of $x$?
• A. ${\log}_{e}|t-2|$
• B. ${\log}_{e}|t-1|$
• C. ${\log}_{e}|t+1|$
• D. ${\log}_{e}|t|$

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$