Second Fundamental Theorem of Calculus

Q.

The domain of definition of the function $y\left(x\right)$ is given by the equation $2^x+2^y=2$ is

1. $0<x\le 1$

2. $0\le x\le 1$

3. $-\infty <x\le 0$

4. $-\infty <x<1$

Q.

If$C_0,C_1,C_2..,C_n$ denote the binomial coefficients in the expansion of

${(1+x)}^n,then{C}_0+(C_1/2)+(C_2/3)+\dots ..C_n/(n+1)=$

1. $[2^{n+1}–1]/(n+1)$

2. $[2^n–1]/(n+1)$

3. $[2^{n-1}–1]/(n-1)$

4. $[2^{n+1}–1]/(n+2)$

Q.

If$y=b\cos \log {\left(\frac{x}{n}\right)}^n$, then $\frac{dy}{dx}$ is equal to ?

1. $-nb\sin \log {\left(\frac{x}{n}\right)}^n$

2. $nb\sin \log {\left(\frac{x}{n}\right)}^n$

3. $\left(\frac{-nb}{x}\right)\sin \log {\left(\frac{x}{n}\right)}^n$

4. none of these

Q.

Find the value of ${\int }_0^2{f}^{}\left(x^2\right)2xdx$

1. $f\left(4\right)–f\left(0\right)$

2. $f\left(4\right)+f\left(0\right)$

3. $f\left(0\right)–f\left(4\right)$

4. None of these

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

1. True
2. False

Q. ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}si{n}^{2}xdx=$

1. $\pi$

2. $\frac{\pi }{2}$

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

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

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

Q. Let $f$ be a real valued function defined as $f\left(x\right)=x^2+x^2\underset{-1}{\overset{1}{\int }}t\cdot f\left(t\right)dt+x^3\underset{-1}{\overset{1}{\int }}f\left(t\right)dt.$ Then which of the following hold(s) good ?

1. $\underset{-1}{\overset{1}{\int }}t\cdot f\left(t\right)dt=\frac{10}{11}$
2. $f\left(1\right)+f(-1)=\frac{30}{11}$
3. $\underset{-1}{\overset{1}{\int }}t\cdot f\left(t\right)dt>\underset{-1}{\overset{1}{\int }}f\left(t\right)dt$
4. $f\left(1\right)-f(-1)=\frac{20}{11}$

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

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

Q. The value of is equal to ${\int }_3^6(\sqrt{x+\sqrt{12x-36}}+\sqrt{x-\sqrt{12x-36}})dx$is equal to

1. $6\sqrt{3}$
2. $4\sqrt{3}$
3. $12\sqrt{3}$
4. $2\sqrt{3}$

Q.

${\int }_0^{\frac{\pi }{2}}{e}^{x}sinxdx=$ [Roorkee 1978]

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

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. If $f"\left(x\right)=kin[0,a],then{\int }_0^{a}f\left(x\right)dx-{\left\{\begin{array}{c}xf\left(x\right)-\frac{x^2}{2!}{f}^{\prime }\left(x\right)+\frac{x^3}{3!}f"\left(x\right)\end{array}\right\}}_0^a$ is

1. $\frac{-k{a}^4}{12}$
2. $\frac{k{a}^4}{24}$
3. $\frac{-k{a}^4}{24}$
4. $\frac{k{a}^4}{12}$

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. If $\underset{0}{\overset{1}{\int }}\frac{e^t}{1+t}dt=a,$ then $\underset{0}{\overset{1}{\int }}\frac{e^t}{(1+t{)}^2}$ is equal to

1. $a-1+\frac{e}{2}$
2. $a+1-\frac{e}{2}$
3. $a+1+\frac{e}{2}$
4. $a+1+\frac{e^2}{2}$

Q. The set of roots of the equations $3x^2-5x-2=0$ can be written in roster form as

1. $\left(-\frac{1}{3},2\right\}$
2. $\left(-2,\frac{1}{3}\right\}$
3. $\left(-2,-\frac{1}{3}\right\}$
4. $(-\frac{1}{3},2)$

Q.

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sqrt{\frac{1}{2}(1-cos2x)}dx=$

1. 0

2. 2

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

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.

The greatest value of $\mathrm{x}+\mathrm{y}+\mathrm{z}$for possible values of $\mathrm{x},\mathrm{y},\mathrm{z}$ such that ${\mathrm{x}}^3+{\mathrm{y}}^3+{\mathrm{z}}^3=1$ is

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

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

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

4. $3^{\frac{4}{3}}$

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.

Find the product of $-\frac{9}{16}{a}^3{b}^{2}and3\frac{1}{5}ab$.

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.

Let f be a positive function. Let
$I_1={\int }_{1-k}^{k}xf\left\{x\right(1-x\left)\right\}dx,I_2={\int }_{1-k}^{k}f\left\{x\right(1-x\left)\right\}dx$
$when2k-1>0.Then\frac{I_1}{I_2}is$ [IIT 1997 Cancelled]

1. 2

2. k

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

4. 1