1
IIT-JEE 1991
Subjective
+4
-0
If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to \infty ,$$ then show that every line $$y=mx$$ IIT-JEE 1991 Mathematics - Definite Integrals and Applications of Integrals Question 30 English
intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$
2
IIT-JEE 1990
Subjective
+4
-0
Prove that for any positive integer $$k$$,
$${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$$
Hence prove that $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} $$
3
IIT-JEE 1990
Subjective
+4
-0
Show that $$\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx} $$
4
IIT-JEE 1990
Subjective
+4
-0
Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$
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