1
AIEEE 2006
+4
-1
$$\int\limits_0^\pi {xf\left( {\sin x} \right)dx}$$ is equal to
A
$$\pi \int\limits_0^\pi {f\left( {\cos x} \right)dx}$$
B
$$\,\pi \int\limits_0^\pi {f\left( {sinx} \right)dx}$$
C
$${\pi \over 2}\int\limits_0^{\pi /2} {f\left( {sinx} \right)dx}$$
D
$$\pi \int\limits_0^{\pi /2} {f\left( {\cos x} \right)dx}$$
2
AIEEE 2006
+4
-1
$$\int\limits_{ - {{3\pi } \over 2}}^{ - {\pi \over 2}} {\left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x + 3\pi } \right)} \right]} dx$$ is equal to
A
$${{{\pi ^4}} \over {32}}$$
B
$${{{\pi ^4}} \over {32}} + {\pi \over 2}$$
C
$${\pi \over 2}$$
D
$${\pi \over 4} - 1$$
3
AIEEE 2006
+4
-1
The value of $$\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1$$ where $${\left[ x \right]}$$ denotes the greatest integer not exceeding $$x$$ is
A
$$af\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( {\left[ a \right]} \right)} \right\}$$
B
$$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( {\left[ a \right]} \right)} \right\}$$
C
$$\left[ a \right]f\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( a \right)} \right\}$$
D
$$af\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( a \right)} \right\}$$
4
AIEEE 2005
+4
-1
Let $$f:R \to R$$ be a differentiable function having $$f\left( 2 \right) = 6$$,
$$f'\left( 2 \right) = \left( {{1 \over {48}}} \right)$$. Then $$\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {{{4{t^3}} \over {x - 2}}dt}$$ equals :
A
$$24$$
B
$$36$$
C
$$12$$
D
$$18$$
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