1
JEE Main 2022 (Online) 29th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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
$$\frac{1}{2}$$
2
JEE Main 2022 (Online) 29th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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
$$\frac{9}{2 \log _{e}(10)}$$
3
JEE Main 2022 (Online) 29th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The area of the region

$$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$$ is equal to :

A
$$\frac{5}{2} \sin ^{-1}\left(\frac{3}{5}\right)-\frac{1}{2}$$
B
$$\frac{5 \pi}{4}-\frac{3}{2}$$
C
$$\frac{3 \pi}{4}+\frac{3}{2}$$
D
$$\frac{5 \pi}{4}-\frac{1}{2}$$
4
JEE Main 2022 (Online) 28th July Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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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