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1

IIT-JEE 2008

MCQ (Single Correct Answer)
Let $$I = \int {{{{e^x}} \over {{e^{4x}} + {e^{2x}} + 1}}dx,\,\,J = \int {{{{e^{ - x}}} \over {{e^{ - 4x}} + {e^{ - 2x}} + 1}}dx.} } $$ Then
for an arbitrary constant $$C$$, the value of $$J -I$$ equals
A
$${1 \over 2}\log \left( {{{{e^{4x}} - {e^{2x}} + 1} \over {{e^{4x}} + {e^{2x}} + 1}}} \right) + C$$
B
$${1 \over 2}\log \left( {{{{e^{2x}} + {e^x} + 1} \over {{e^{2x}} - {e^x} + 1}}} \right) + C$$
C
$${1 \over 2}\log \left( {{{{e^{2x}} - {e^x} + 1} \over {{e^{2x}} + {e^x} + 1}}} \right) + C$$
D
$${1 \over 2}\log \left( {{{{e^{4x}} + {e^{2x}} + 1} \over {{e^{4x}} - {e^{2x}} + 1}}} \right) + C$$
2

IIT-JEE 2007

MCQ (Single Correct Answer)
Let $$F(x)$$ be an indefinite integral of $$si{n^2}x.$$

STATEMENT-1: The function $$F(x)$$ satisfies $$F\left( {x + \pi } \right) = F\left( x \right)$$
for all real $$x$$. because

STATEMENT-2: $${\sin ^2}\left( {x + \pi } \right) = {\sin ^2}x$$ for all real $$x$$.

A
Statement-1 is True, Statement-2 is True; Statement-2 is is a correct explanation for Statement-1.
B
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
C
Statement- 1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.
3

IIT-JEE 2006

MCQ (Single Correct Answer)
$$\int {{{{x^2} - 1} \over {{x^3}\sqrt {2{x^4} - 2{x^2} + 1} }}dx = } $$
A
$${{\sqrt {2{x^4} - 2{x^2} + 1} } \over {{x^2}}} + c$$
B
$${{\sqrt {2{x^4} - 2{x^2} + 1} } \over {{x^3}}} + c$$
C
$${{\sqrt {2{x^4} - 2{x^2} + 1} } \over {{x}}} + c$$
D
$${{\sqrt {2{x^4} - 2{x^2} + 1} } \over {{2x^2}}} + c$$
4

IIT-JEE 2005 Screening

MCQ (Single Correct Answer)
If $$\int\limits_{\sin x}^1 {{t^2}f\left( t \right)dt = 1 - \sin x,} $$ then f$$\left( {{1 \over {\sqrt 3 }}} \right)$$ is
A
$${1 \over 3}$$
B
$${{1 \over {\sqrt 3 }}}$$
C
$$3$$
D
$${\sqrt 3 }$$

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