1
IIT-JEE 1997
Subjective
+5
-0
Let $$a+b=4$$, where $$a<2,$$ and let $$g(x)$$ be a differentiable function.

If $${{dg} \over {dx}} > 0$$ for all $$x$$, prove that $$\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} } $$
increases as $$(b-a)$$ increases.

2
IIT-JEE 1997
Fill in the Blanks
+2
-0
The value of $$\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx} $$ is ...............
3
IIT-JEE 1997
Fill in the Blanks
+2
-0
Let $${d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.$$ If $$\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \right) - F\left( 1 \right)$$
then one of the possible values of $$k$$ is ............
4
IIT-JEE 1997
MCQ (Single Correct Answer)
+2
-0.5
If $$g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,} $$ then $$g\left( {x + \pi } \right)$$ equals
A
$$g\left( x \right) + g\left( \pi \right)$$
B
$$g\left( x \right) - g\left( \pi \right)$$
C
$$g\left( x \right) g\left( \pi \right)$$
D
$${{g\left( x \right)} \over {g\left( \pi \right)}}$$
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