AIEEE 2007 | Definite Integration Question 244 | Mathematics

Let $$I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx} $$ and $$J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .$$ Then which one of the following is true?

$$1 > {2 \over 3}$$ and $$J > 2$$

$$1 < {2 \over 3}$$ and $$J < 2$$

$$1 < {2 \over 3}$$ and $$J > 2$$

$$1 > {2 \over 3}$$ and $$J < 2$$

AIEEE 2006

MCQ (Single Correct Answer)

-1

$$\int\limits_0^\pi {xf\left( {\sin x} \right)dx} $$ is equal to

$$\pi \int\limits_0^\pi {f\left( {\cos x} \right)dx} $$

$$\,\pi \int\limits_0^\pi {f\left( {sinx} \right)dx} $$

$${\pi \over 2}\int\limits_0^{\pi /2} {f\left( {sinx} \right)dx} $$

$$\pi \int\limits_0^{\pi /2} {f\left( {\cos x} \right)dx} $$

AIEEE 2006

MCQ (Single Correct Answer)

-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

$${{{\pi ^4}} \over {32}}$$

$${{{\pi ^4}} \over {32}} + {\pi \over 2}$$

$${\pi \over 2}$$

$${\pi \over 4} - 1$$

AIEEE 2006

MCQ (Single Correct Answer)

-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

$$af\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( {\left[ a \right]} \right)} \right\}$$

$$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( {\left[ a \right]} \right)} \right\}$$

$$\left[ a \right]f\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( a \right)} \right\}$$

$$af\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( a \right)} \right\}$$

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