Definite Integration - Calculus - Mathematics Previous Year Questions

If $[x]$ denotes the greatest integer function of $x$ and

$$\int_{-3 / 2}^{3 / 2}[2 x-3] d x=k, \text { then }\left|k+\frac{1}{2}\right|=$$

$$\int_1^2 x \sqrt{4-x^2} d x=$$

Let $I=\int_{-\pi / 4}^{\pi / 4} \frac{1}{2-\cos 2 x}\left(\frac{\beta}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. Given that $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ and $\tan ^{-1}(\sqrt{3})=\pi / 3$. Then, $3 I^2=$

$$\int_1^4\left(x+\sqrt{x}+\frac{1}{x}\right) d x-\int_1^{2 \log 2} d x=$$

$$\int\limits_2^5 \sqrt{\frac{5-x}{x-2}} d x=$$

$$\int\limits_0^{\frac{\pi}{2}} \sin ^6 x \cos ^4 x d x=$$

$$\int_0^{\pi / 4} \frac{\sec x}{1+2 \sin ^2 x} d x=$$

$$\lim\limits_{n \rightarrow \infty}\left[\frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\ldots \ldots+\frac{1}{n} \sec ^2 1\right]=$$

$$\int_0^\pi \frac{x \cos ^2 x}{1+\sin x} d x=$$

If $[x]$ represents greatest integer function, then

$$\int_{-2}^2[2-x] d x=$$

$$\int_0^2 x^3(2-x)^4 d x=$$

$$\int_0^2 \frac{x}{(2-x)^{\frac{3}{4}}} d x=$$

$$\int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x=$$

$$\int_0^3\left|x^2-3 x+2\right| d x=$$

$$\int_0^2 x^{\frac{5}{2}} \sqrt{2-x} d x=$$

$$\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots(2)\right]^{1 / n}=$$

$$\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x=$$

$$\int_0^{\pi / 2} \frac{x \tan x \sec ^2 x}{\tan ^4 x+1} d x=$$

If $\int_0^3\left(3 x^2-4 x+2\right) d x=k$, then an integer root of $3 x^2-4 x+2=3 k / 5$ is

$$\int_{-1}^1 x|x| d x=$$