Second Fundamental Theorem of Calculus

Q.

${\int }_0^1{e}^{2Inx}dx=$ [MP PET 1990]

1. 0
2. 
3. 

4. 
   

Q.

$Iff\left(x\right)={\int }_{-1}^x|t|dt,x\ge -1,then$ [MNR 1994]

1. 
2. 
3. 

4. 
   

Q.

The numbers P, Q and R for which the function
$f\left(x\right)=P{e}^{2x}+Q{e}^x+Rx$ satisfies the conditions
$f\left(0\right)=-1,f^{{}^{\prime }}\left(log2\right)=31$ and ${\int }_0^{log4}\left[f\right(x)-Rx]dx=\frac{39}{2}$ are
given by

1. 
   

2. 
   

3. 
   

4. 
   

Q. ${\int }_1^t\frac{e^x}{x}(1+xlogx)dx=$

1. 
2. 
3. 
4. None of these

Q. Value of ${\int }_0^5(\sqrt{x+2\sqrt{x+1}}+\sqrt{x-2\sqrt{x-1}})dx$ is

1. $\frac{16}{3}$
2. $\frac{8}{3}$
3. $\frac{32}{3}$
4. $\frac{34}{3}$

Q. Let $f\left(x\right)=\frac{1}{2}{a}_0+{\sum }_{i-1}^n{a}_{i}cos\left(ix\right)+{\sum }_{j-1}^n{b}_{j}sin\left(jx\right),then{\int }_{-\pi }^{\pi }f\left(x\right)coskxdx$ then is equal to

1. $a_k$
2. $b_k$
3. $\pi a_k$
4. $\pi b_k$

Q.

${\int }_0^{\frac{\pi }{4}}ta{n}^{2}xdx=$ [Roorkee 1983, Pb. CET 2000]

1. 
2. 
3. 
4. 

Q. ${\int }_0^{\frac{\pi }{2}}co{s}^{2}xdx=$

1. 
   

2. 
   

3. 
   

4. 
   

Q.

Let f(x) be a function satisfying $f^{{}^{\prime }}\left(x\right)=f\left(x\right)$ with f(0) = 1
and g(x) be the function satisfying $f\left(x\right)+g\left(x\right)=x^2$
The value of integral ${\int }_0^{1}f\left(x\right)g\left(x\right)dx$ is equal to [AIEEE 2003; DCE 2005]

1. 
   

2. 
3. 

4. None of these

Q. Second fundamental theorem of integral calculus gives the method of calculating Definite Integral.

1. True
2. False