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1

IIT-JEE 1987

Subjective
Evaluate :$$\,\,\int {\left[ {{{{{\left( {\cos 2x} \right)}^{1/2}}} \over {\sin x}}} \right]dx} $$

Answer

$${1 \over {\sqrt 2 }}\,\log \left[ {{{\sqrt 2 + \sqrt {1 - {{\tan }^2}x} } \over {\sqrt 2 - \sqrt {1 - {{\tan }^2}x} }}} \right] - \log \left( {\cot x + \sqrt {{{\cot }^2}x - 1} } \right) + C$$
2

IIT-JEE 1985

Subjective
Evaluate the following $$\int {\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}dx} } $$

Answer

$$ - 2\sqrt {1 - x} + {\cos ^{ - 1}}\sqrt x + \sqrt x \sqrt {1 - x} + C$$
3

IIT-JEE 1984

Subjective
Evaluate the following $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $$

Answer

$$ - {\left( {1 + {1 \over {{x^{ \to 4}}}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}}} + C$$
4

IIT-JEE 1983

Subjective
Evaluate : $$\int {{{\left( {x - 1} \right){e^x}} \over {{{\left( {x + 1} \right)}^3}}}dx} $$

Answer

$${{{e^x}} \over {{{\left( {x + 1} \right)}^2}}} + C$$

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