Second Fundamental Theorem of Calculus

Q. For a positive constant $t,$ let $\alpha ,\beta$ be the roots of the quadratic equation $x^2+t^{2}x-2t=0$. If the minimum value of $\underset{-1}{\overset{2}{\int }}[(x+\frac{1}{{\alpha }^2})(x+\frac{1}{{\beta }^2})+\frac{1}{\alpha \beta }]dx$ is $\sqrt{\frac{a}{b}}+c$ where $a,b,c\in N,$ then the least possible value of $(a+b+c)$ is

Q. ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}cosec2xdx=$

1. log 3
2. log $\sqrt{3}$

3. log 9

4. None of these

Q. The integral $\underset{\frac{\pi }{4}}{\overset{\frac{3\pi }{4}}{\int }}\frac{dx}{1+\cos x}$ is equal to:

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

Q. ${\int }_1^2\sqrt{(x-1)(2-x)}dx=$

1. $\frac{\pi }{8}$
2. $\frac{\pi }{4}$
3. $\frac{1}{8}$
4. none

Q. If $z\ne 0$ then ${\int }_0^{100}\left[arg|z|\right]dx$ is (where [.] denotes the greatest integer function)

1. \N
2. 10
3. 100
4. not defined

Q. Let $f\left(x\right)$ and $\varphi \left(x\right)$ are two continuous functions on $R$ satisfying $\varphi \left(x\right)=\underset{a}{\overset{x}{\int }}f\left(t\right)dt,a\ne 0$. If $f\left(x\right)$ is an even function, then which of the following statements are correct?

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. P = 2, Q = -3, R = 4

2. P = -5, Q = 2, R = 3

3. P = 5, Q = -2, R = 3

4. P = 5, Q = -6, R = 3

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

1. True
2. False

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. $\frac{1}{4}(e-7)$

2. $\frac{1}{4}(e-2)$
3. $\frac{1}{2}(e-3)$

4. $Noneofthese$

Q. ${\int }_a^b\frac{logx}{x}dx=$ [MP PET 1994]

1. $log\left(\frac{logb}{loga}\right)$
2. $log\left(ab\right)log\left(\frac{b}{a}\right)$
3. $\frac{1}{2}log\left(ab\right)log\left(\frac{b}{a}\right)$
4. $\frac{1}{2}log\left(ab\right)log\left(\frac{a}{b}\right)$

Q. The value of $\sum _{n=1}^{100}\underset{n-1}{\overset{n}{\int }}{e}^{x-\left[x\right]}dx,$ where $\left[x\right]$ is the greatest integer $\le x,$ is :

1. $100(e-1)$
2. $100e$
3. $100(1-e)$
4. $100(1+e)$

Q. ${\int }_1^2{e}^x(\frac{1}{x}-\frac{1}{x^2})dx=$ [MNR 1990; AMU 1999; UPSEAT 2000; Pb. CET 2004]

1. $\frac{e^2}{2}+e$

2. $e-\frac{e^2}{2}$
3. $\frac{e^2}{2}-e$

4. $Noneofthese$

Q. $\begin{array}{cc}\text{List I}& \text{List II}\\ \text{(I)}{\int }_0^2\frac{x^4+1}{(x^2+2{)}^{\frac{3}{4}}}dx=& \text{(P)}\sqrt{3\sqrt{2}}\\ \text{(II)}{\int }_1^3(\sqrt{1+(x-1{)}^3}+\sqrt[3]{3x^2-1})dx=& \text{(Q)}\sqrt{2\sqrt{3}}\\ \text{(III)}{\int }_0^2\sqrt[3]{3x^3-3x^2+8x-6}dx=& \text{(R)}6\\ \text{(IV)}{\int }_e^{\pi }\frac{2^{\frac{x}{e}}-2^{\frac{\pi }{x}}}{x}dx=& \text{(S)}0\\ & \text{(T)}\frac{\pi -e}{2}\end{array}$
Which of the following is only INCORRECT combination?

1. (I)-(P)
2. (IV)-(S)
3. (II)-(S)
4. (III)-(S)

Q.

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

1. f and f' are continous for x + 1 > 0
2. f is continous but f' is not continous for x + 1 > 0
3. f and f' are not continous at x = 0
4. f is continous at x = 0 but f' is not so

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

1. $e^t$
2. $e^t-e$
3. $e^t+e$
4. $Noneofthese$

Q. Let $I_n={\int }_0^{\infty }{e}^{-x}(sinx{)}^{n}dx,nϵN,n>1$ then $\frac{I_{2008}}{I_{2006}}$ equals

1. $\frac{2007\times 2006}{{2008}^2+1}$
2. $\frac{2008\times 2007}{{2008}^2+1}$
3. $\frac{2006\times 2004}{{2008}^2-1}$
4. $\frac{2008\times 2007}{{2008}^2-1}$

Q. $f\left(x\right)$ satisfies the relation $f\left(x\right)-\lambda \underset{0}{\overset{\pi /2}{\int }}\sin x\cos tf\left(t\right)dt=\sin x.$

If $\lambda >2,$ then $f\left(x\right)$ decreases in which of the following interval?

1. $(0,\pi )$
2. $(\frac{\pi }{2},\frac{3\pi }{2})$
3. $(-\frac{\pi }{2},\frac{\pi }{2})$
4. none of these

Q. ${\int }_0^{\pi }sinx$ dx is equal to -

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

Q. The value of $\underset{1}{\overset{e}{\int }}\frac{x^3+1}{x^4+x^3+x^2+x}dx$ is

1. $\frac{4+\pi }{4}-{\cot }^{-1}\left(\sqrt{1+e^2}\right)$
2. $\frac{4+\pi }{4}-{\sec }^{-1}\left(\sqrt{1+e^2}\right)$
3. $\frac{4-\pi }{4}-{\cos }^{-1}\left(\sqrt{1+e^2}\right)$
4. $\frac{4-\pi }{4}-{\sec }^{-1}\left(\sqrt{1+e^2}\right)$

Q.

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

1. $0$
2. $\frac{1}{2}$
3. $\frac{1}{3}$

4. $\frac{1}{4}$

Q. ${\int }_a^b\frac{logx}{x}dx=$ [MP PET 1994]

1. $log\left(\frac{logb}{loga}\right)$
2. $log\left(ab\right)log\left(\frac{b}{a}\right)$
3. $\frac{1}{2}log\left(ab\right)log\left(\frac{b}{a}\right)$
4. $\frac{1}{2}log\left(ab\right)log\left(\frac{a}{b}\right)$

Q. The value of $\sum _{n=1}^{100}\underset{n-1}{\overset{n}{\int }}{e}^{x-\left[x\right]}dx,$ where $\left[x\right]$ is the greatest integer $\le x,$ is :

1. $100(e-1)$
2. $100e$
3. $100(1-e)$
4. $100(1+e)$

Q. Find $\underset{0}{\overset{2}{\int }}{x}^{2}dx$ as the limit of a sum

1. $4$
2. $8$
3. $\frac{4}{3}$
4. $\frac{8}{3}$

Q.

${\int }_0^{\frac{x}{2}}\frac{x+sinx}{1+cosx}dx=$ [Roorkee 1978]

1. -log 2
2. log 2
3. $\frac{\pi }{2}$
4. \N

Q. $\text{Let}\lambda ={\int }_1^9(\frac{1}{2}+\sqrt{\frac{1}{4}+{\log }_{3}x})dx+{\int }_1^2\frac{3^{x^2-\left[x\right]}}{3^{\left\{x\right\}}}dx$ then $2\lambda \text{equals}$ [where [.] denotes the greatest integer function and {.} denotes the fractional part function]