MCQ
$\int\limits_1^e {\left( {\frac{{{{\tan }^{ - 1}}x}}{x} + \frac{{\ln x}}{{\left( {1 + {x^2}} \right)}}} \right)} \,dx$ is equal to
- A$\frac{1}{e}{\tan ^{ - 1}}e$
- ✓$tan^{-1}e$
- C$e\ {\tan ^{ - 1}}\left( {\frac{1}{e}} \right)$
- D$tan^{-1}(lne)$
$=\left(\tan ^{-1} x . \ln x\right)_{1}^{e}-\int_{1}^{e} \frac{1}{\left(1+x^{2}\right)} \ln x d x+\int_{1}^{e} \frac{\ln x}{\left(1+x^{2}\right)} d x$
$ = {\tan ^{ - 1}}(e) \cdot \ln e - {\tan ^{ - 1}}(1) \cdot \ln 1$
$=\tan ^{-1}(e)$
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$(A)$ $\int_0^1 x \cos x d x \geq \frac{3}{8}$ $(B)$ $\int_0^1 x \sin x d x \geq \frac{3}{10}$ $(C)$ $\int_0^1 x^2 \cos x d x \geq \frac{1}{2}$ $(D)$ $\int_0^1 x^2 \sin x d x \geq \frac{2}{9}$
| List $I$ | List $II$ |
| $P.$ The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\leq 2$, satisfying $f(0)=0$ and $\int_0^1 f(x) d x=1$, is | $1.$ $8$ |
| $Q.$ The number of points in the interval $\mid-\sqrt{13}, \sqrt{13})$ at which $f(x)=\sin \left(x^2\right)+\cos \left(x^2\right)$ attains its maximum value, is | $2.$ $2$ |
| $R.$ $\int_{-2}^2 \frac{3 x^2}{\left(1+e^x\right)} d x$ equals | $3.$ $4$ |
| $S.$ $\frac{\left(\int_{-1 / 2}^{1 / 2} \cos 2 x \log \left(\frac{1+x}{1-x}\right) d x\right)}{\left(\int_0^{1 / 2} \cos 2 x \log \left(\frac{1+x}{1-x}\right) d x\right)}$ equals | $4.$ $0$ |
Codes: $ \quad P \quad Q \quad R \quad S$