AIEEE 2007 | Definite Integration Question 251 | Mathematics

Let $$F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),$$ where $$f\left( x \right) = \int\limits_l^x {{{\log t} \over {1 + t}}dt,} $$ Then $$F(e)$$ equals

$$1$$

$$2$$

$$1/2$$

$$0$$

AIEEE 2007

MCQ (Single Correct Answer)

-1

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

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

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

$${\pi \over 2}$$

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

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