Integration of Piecewise Continuous Functions

Q. Consider the integral $I=\underset{0}{\overset{10}{\int }}\frac{\left[x\right]e^{\left[x\right]}}{e^{x-1}}dx,$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x.$ Then the value of $I$ is equal to :

1. $45(e-1)$
2. $45(e+1)$
3. $9(e-1)$
4. $9(e+1)$

Q.

${\int }_0^2\left(\right[x{]}^2-\left[x^2\right])dx$ is equal to

1. $-4+\sqrt{3}+\sqrt{2}$
   

2. $4+\sqrt{3}-\sqrt{2}$
   

3. $4+2\sqrt{3}+\sqrt{2}$
   

4. None of these

Q. The value of the integral, $\underset{1}{\overset{3}{\int }}[x^2-2x-2]dx,$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ is

1. $-4$
2. $-5$
3. $-\sqrt{2}-\sqrt{3}-1$
4. $-\sqrt{2}-\sqrt{3}+1$

Q. Let f(x) = x - [x], for every real number x, where [x] is the greatest integer less than or equal to x. Then, the value of ${\int }_{-1}^{1}f\left(x\right)\mathrm{dx}$ is

1. 1
2. \N
3. $\frac{1}{2}$
4. $\frac{3}{2}$

Q. ${\int }_0^{\frac{\pi }{2}}\frac{coxx-sinx}{1+sinxcosx}dx=$

1. 2
2. -2
3. \N

4. None of these

Q. Find the value of ${\int }_0^{\infty }\left[\frac{2}{e^x}\right]$

1. \N
2. ln2
3. 1
4. 2

Q. If $f:R\to R$ defined by $f\left(x\right)=\{\begin{array}{c}2x+5,x>0\\ 3x-2,x\le 0\end{array}$ then $f$ is

1. into function
2. one-one function
3. onto function
4. many-one function

Q. Let $f$ be a non-negative continuous and twice differentiable function defined on $R$ such that $f\left(x\right)+f(x+\frac{3}{2})=6,\forall x\in R.$ Then

1. $f\left(x\right)$ is a periodic function
2. $f\left(x\right)$ is a non-periodic function
3. $\underset{0}{\overset{300}{\int }}f\left(x\right)dx=900$
4. If $f\left(0\right)=3,$ then $f^{\prime \prime }\left(x\right)$ has at least $3$ roots in $(0,6)$

Q.

If $f\left(x\right)=\frac{e^x}{1+e^x},I_1={\int }_{f(-a)}^{f\left(a\right)}xg\left\{x\right(1-x\left)\right\}dx$, and $I_2={\int }_{f(-a)}^{f\left(a\right)}g\left\{x\right(1-x\left)\right\}dx$, then the value of $\frac{I_2}{I_1}$ [AIEEE 2004]

1. 1
2. -3
3. -1
4. 2

Q. ${\int }_0^{\pi }xf\left(sinx\right)dx=$

1. $\pi {\int }_0^{\pi }f\left(sinx\right)dx$
2. $\frac{\pi }{2}{\int }_0^{\pi }f\left(sinx\right)dx$
3. $\frac{\pi }{2}{\int }_0^{\frac{\pi }{2}}f\left(sinx\right)dx$
4. $Noneofthese$

Q. The value of $\underset{-2}{\overset{2}{\int }}|3x^2-3x-6|dx$ is

Q.

${\int }_0^{\frac{\pi }{2}}\frac{\sqrt{cotx}}{\sqrt{cotx}+\sqrt{tanx}}dx=$ [MP PET 1990, 95; IIT 1983; MNR 1990]

1. $\pi$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{3}$

Q.

Let $f\left(x\right)=max(x+\left|x\right|,x-\left[x\right])$ where [x] = the greatest integer in $x\le x$. Then ${\int }_{-2}^{2}f\left(x\right)dx$ is equal to

1. 3

2. 2

3. 1

4. None of these

Q. If f(a+b-x) = f(x), then ${\int }_a^{b}xf\left(x\right)dx$ is equal to

1. $\frac{a+b}{2}{\int }_a^{b}f(b-x)dx$
2. $\frac{a+b}{2}{\int }_a^{b}f\left(x\right)dx$
3. $\frac{b-a}{2}{\int }_a^{b}f\left(x\right)dx$
4. $\frac{a+b}{2}{\int }_a^{b}f(a+b+x)dx$

Q. A function $f:R\to R^{+}$ satisfies $f(x+y)=f\left(x\right)f\left(y\right)\forall x,y\in R.$ If $f\left(0\right)=1$ and $f^{\prime }\left(0\right)=2$, then the value of $\underset{0}{\overset{{\log }_{e}3}{\int }}\left[f\right(x\left)e^{-x}\right]dx$ is

(where $[\cdot ]$ is represents the greatest integer function)

1. ${\log }_e\left(\frac{3}{2}\right)$
2. ${\log }_e\left(\frac{9}{2}\right)$
3. ${\log }_e\left(\frac{9}{4}\right)$
4. ${\log }_{e}3$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=\{\begin{array}{c}\left[x\right],x\le 2\\ 0,x>2\end{array},$
where $\left[x\right]$ is the greatest integer less than or equal to $x.$ If $I=\underset{-1}{\overset{2}{\int }}\frac{xf\left(x^2\right)}{2+f(x+1)}dx,$ then the value of $(4I–1)$ is

Q. If $I_n=\underset{\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}{\cot }^{n}xdx$, then

1. $\frac{1}{I_2+I_4},\frac{1}{I_3+I_5},\frac{1}{I_4+I_6}\text{are in G.P.}$
2. $\frac{1}{I_2+I_4},\frac{1}{I_3+I_5},\frac{1}{I_4+I_6}\text{are in A.P.}$
3. $I_2+I_4,I_3+I_5,I_4+I_6\text{are in A.P.}$
4. $I_2+I_4,(I_3+I_5{)}^2,I_4+I_6\text{are in G.P.}$

Q. ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{cosx}{1+e^x}dx=$

1. \N
2. -2
3. 2
4. 1