Question 1: Palette 1/6

BITSAT 2023 MCQ

The value of integral $\int \frac{d x}{(1+x)^{3 / 4}(x-2)^{5 / 4}}$ is is equal to

A

$-\frac{3}{4}\left(\frac{x+1}{x-2}\right)^{1 / 4}+C$

B

$-\frac{3}{4}\left(\frac{x-2}{x+1}\right)^{1 / 4}+C$

C

$-\frac{4}{3}\left(\frac{x+1}{x-2}\right)^{1 / 4}+C$

D

$-\frac{4}{3}\left(\frac{x-2}{x+1}\right)^{1 / 4}+C$

Question 2: Palette 2/6

BITSAT 2023 MCQ

Let $f(x)=\int \frac{\sqrt{x}}{(1+x)^2} d x$, where $x \geq 0$. Then, $f(3)-f(1)$ is equal to

A

$\frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}$

B

$-\frac{\pi}{6}+\frac{1}{2}+\frac{\sqrt{3}}{4}$

C

$-\frac{\pi}{12}+\frac{1}{2}+\frac{\sqrt{3}}{4}$

D

$\frac{\pi}{6}+\frac{1}{2}-\frac{\sqrt{3}}{4}$

Question 3: Palette 3/6

BITSAT 2022 MCQ

The value of $\int {{1 \over {{{[{{(x - 1)}^3}{{(x + 2)}^5}]}^{{1 \over 4}}}}}dx} $, is

A

${4 \over 3}{\left( {{{x + 1} \over {x - 2}}} \right)^{{1 \over 4}}} + C$

B

${3 \over 4}{\left( {{{x - 1} \over {x + 2}}} \right)^{{1 \over 4}}} + C$

C

${4 \over 3}{\left( {{{x - 1} \over {x + 2}}} \right)^{{1 \over 4}}} + C$

D

${1 \over 3}{\left( {{{2x - 1} \over {4x - 3}}} \right)^{{1 \over 4}}} + C$

Question 4: Palette 4/6

BITSAT 2022 MCQ

Let $f(x) = \int {{{{x^2}dx} \over {(1 + {x^2})(1 + \sqrt {1 + {x^2}} )}}} $ and $f(0) = 0$, then the value of $f(1)$ be

A

$\log (1 + \sqrt 2 )$

B

$\log (1 + \sqrt 2 ) - {\pi \over 4}$

C

$\log (1 + \sqrt 2 ) + {\pi \over 2}$

D

None of these

Question 5: Palette 5/6

BITSAT 2021 MCQ

$\int {{1 \over {1 - 2\sin x}}dx} $ is equal to

A

${1 \over {2\sqrt 3 }}\log \left| {{{\tan {x \over 2} - 2 - \sqrt 3 } \over {\tan {x \over 2} - 2 + \sqrt 3 }}} \right| + c$

B

${{\sqrt 3 } \over 2}\log \left| {{{\tan {x \over 2} - 2 - \sqrt 3 } \over {\tan {x \over 2} - 2 + \sqrt 3 }}} \right| + c$

C

${1 \over {\sqrt 3 }}\log \left| {{{\tan {x \over 2} - 2 - \sqrt 3 } \over {\tan {x \over 2} - 2 + \sqrt 3 }}} \right| + c$

D

None of the above

Question 6: Palette 6/6

BITSAT 2020 MCQ

$\int {{{8{x^{43}} + 13{x^{38}}} \over {{{({x^{13}} + {x^5} + 1)}^4}}}dx} $ equals to

A

${{{x^{39}}} \over {3{{({x^{13}} + {x^5} + 1)}^3}}} + C$

B

${{{x^{39}}} \over {{{({x^{13}} + {x^5} + 1)}^3}}} + C$

C

${{{x^{39}}} \over {5{{({x^{13}} + {x^5} + 1)}^3}}} + C$

D

None of these