1
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

If $$[t]$$ denotes the greatest integer $$\leq t$$, then the value of $$\int_{0}^{1}\left[2 x-\left|3 x^{2}-5 x+2\right|+1\right] \mathrm{d} x$$ is :

A
$$\frac{\sqrt{37}+\sqrt{13}-4}{6}$$
B
$$\frac{\sqrt{37}-\sqrt{13}-4}{6}$$
C
$$\frac{-\sqrt{37}-\sqrt{13}+4}{6}$$
D
$$\frac{-\sqrt{37}+\sqrt{13}+4}{6}$$
2
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

The integral $$\int\limits_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} \mathrm{~d} x$$ is equal to :

A
$$\tan ^{-1}(2)$$
B
$$\tan ^{-1}(2)-\frac{\pi}{4}$$
C
$$\frac{1}{2} \tan ^{-1}(2)-\frac{\pi}{8}$$
D
$$\frac{1}{2}$$
3
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

If $$f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$$, then $$f\left(\mathrm{e}^{3}\right)+f\left(\mathrm{e}^{-3}\right)$$ is equal to :

A
9
B
$$\frac{9}{2}$$
C
$$\frac{9}{\log _{e}(10)}$$
D
$$\frac{9}{2 \log _{e}(10)}$$
4
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

Let $$I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$$ Then :

A
$$50 I_{6}-9 I_{5}=x I_{5}^{\prime}$$
B
$$50 I_{6}-11 I_{5}=x I_{5}^{\prime}$$
C
$$50 I_{6}-9 I_{5}=I_{5}^{\prime}$$
D
$$50 I_{6}-11 I_{5}=I_{5}^{\prime}$$
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