1
IIT-JEE 2007
Subjective
+6
-0
Match the integrals in Column $$I$$ with the values in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.

Column $$I$$
(A) $$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}}$$
(B) $$\int\limits_0^1 {{{dx} \over {\sqrt {1 - {x^2}} }}}$$
(C) $$\int\limits_2^3 {{{dx} \over {1 - {x^2}}}}$$
(D) $$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}}$$

Column $$II$$
(p) $${1 \over 2}\log \left( {{2 \over 3}} \right)$$
(q) $$2\log \left( {{2 \over 3}} \right)$$
(r) $${{\pi \over 3}}$$
(s) $${{\pi \over 2}}$$

2
IIT-JEE 2006
Subjective
+6
-0
The value of $$5050{{\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx} \over {\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx}}$$ is.
3
IIT-JEE 2005
Subjective
+2
-0
Evaluate $$\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$$
4
IIT-JEE 2004
Subjective
+2
-0
If $$y\left( x \right) = \int\limits_{{x^2}/16}^{{x^2}} {{{\cos x\cos \sqrt \theta } \over {1 + {{\sin }^2}\sqrt \theta }}d\theta ,}$$ then find $${{dy} \over {dx}}$$ at $$x = \pi$$
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