Integral Calculus & Area · NDA previous year questions

For the next three (03) items that follow : The line $2y=3x+12$ cuts the parabola $4y=3x^2$. What is the area enclosed by the parabola and the line?

1. A.27 square unit
2. B.36 square unit
3. C.48 square unit
4. D.54 square unit

For the next three (03) items that follow : The line $2y=3x+12$ cuts the parabola $4y=3x^2$. What is the area enclosed by the parabola, the line and the y-axis in the first quadrant?

1. A.7 square unit
2. B.14 square unit
3. C.20 square unit
4. D.21 square unit

For the next two (02) items that follow: Consider the integrals $I_1={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{dx}{1+\sqrt{\tan x}}$ and $I_2={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{\sqrt{\sin x}dx}{\sqrt{\sin x}+\sqrt{\cos x}}$. What is $I_1-I_2$ equal to?

1. A.0
2. B.$2I_1$
3. C.$\pi$
4. D.None of the above

For the next two (02) items that follow: Consider the integrals $I_1={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{dx}{1+\sqrt{\tan x}}$ and $I_2={\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{\sqrt{\sin x}dx}{\sqrt{\sin x}+\sqrt{\cos x}}$. What is $I_1$ equal to?

1. A.$\pi /24$
2. B.$\pi /18$
3. C.$\pi /12$
4. D.$\pi /6$

What is the area bounded by the lines $x=0,y=0$ and $x+y+2=0$ ?

1. A.$\frac{1}{2}$ square unit
2. B.1 square unit
3. C.2 square units
4. D.4 square units

What is the area bounded by $y=\tan x,y=0$ and $x=\frac{\pi }{4}$ ?

1. A.$\ln 2$ square units
2. B.$\frac{\ln 2}{2}$ square units
3. C.$2(\ln 2)$ square units
4. D.None of the above

What is $\int (x\cos x+\sin x)dx$ equal to? where $c$ is an arbitrary constant.

1. A.$x\sin x+c$
2. B.$x\cos x+c$
3. C.$-x\sin x+c$
4. D.$-x\cos x+c$

What is ${\int }_0^2{e}^{\ln x}dx$ equal to?

1. A.1
2. B.2
3. C.4
4. D.None of the above

What is $\int e^{\ln x}dx$ equal to ? where c is the constant of integration.

1. A.$x{e}^{\ln x}+c$
2. B.$-x{e}^{\ln x}+c$
3. C.$x+c$
4. D.$\frac{x^2}{2}+c$

What is ${\int }_{-\pi /6}^{\pi /6}\frac{{\sin }^{5}x{\cos }^{3}x}{x^4}dx$ equal to ?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{\pi }{8}$
4. D.0

What is $\int \frac{dx}{x\ln x}$ equal to ? where c is the constant of integration.

1. A.$\ln (\ln x)+c$
2. B.$\ln x+c$
3. C.$(\ln x{)}^2+c$
4. D.None of the above

What is the area of the region enclosed by $y=2|x|$ and $y=4$ ?

1. A.2 square unit
2. B.4 square unit
3. C.8 square unit
4. D.16 square unit

What is $\int a^x{e}^{x}dx$ equal to? where c is the constant of integration

1. A.$\frac{a^x{e}^x}{\ln a}+c$
2. B.$a^x{e}^x+c$
3. C.$\frac{a^x{e}^x}{\ln (ae)}+c$
4. D.None of the above

What is ${\int }_{-1}^{1}x|x|dx$ equal to?

1. A.2
2. B.1
3. C.0
4. D.$-1$

What is ${\int }_0^1\frac{{\tan }^{-1}x}{1+x^2}dx$ equal to?

1. A.$\frac{{\pi }^2}{8}$
2. B.$\frac{{\pi }^2}{32}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{8}$

What is ${\int }_0^{\pi /2}\sin 2x\ln (\cot x)dx$ equal to?

1. A.0
2. B.$\pi \ln 2$
3. C.$-\pi \ln 2$
4. D.$\frac{\pi \ln 2}{2}$

What is the area of the portion of the curve $y=\sin x$, lying between $x=0,y=0$ and $x=2\pi$?

1. A.1 square unit
2. B.2 square units
3. C.4 square units
4. D.8 square units

What is $\int \frac{\ln x}{x}dx$ equal to? where c is the constant of integration

1. A.$\frac{(\ln x{)}^2}{2}+c$
2. B.$\frac{(\ln x)}{2}+c$
3. C.$(\ln x{)}^2+c$
4. D.None of the above

What is the area of the region bounded by the lines $y=x,y=0$ and $x=4$?

1. A.4 square units
2. B.8 square units
3. C.12 square units
4. D.16 square units

What is $\int \left(\frac{1}{{\cos }^{2}x}-\frac{1}{{\sin }^{2}x}\right)dx$ equal to? where c is the constant of integration

1. A.$2\mathrm{cosec}2x+c$
2. B.$-2\cot 2x+c$
3. C.$2\sec 2x+c$
4. D.$-2\tan 2x+c$

What is $\int (x^2+1{)}^{\frac{5}{2}}xdx$ equal to? where c is a constant of integration.

1. A.$(x^2+1{)}^{\frac{7}{2}}+c$
2. B.$\frac{2}{7}(x^2+1{)}^{\frac{7}{2}}+c$
3. C.$\frac{1}{7}(x^2+1{)}^{\frac{7}{2}}+c$
4. D.None of the above

If $f(x)=x^2-6x+8$ and there exists a point c in the interval $[2,4]$ such that $f^{\prime }(c)=0$, then what is the value of c?

1. A.2.5
2. B.2.8
3. C.3
4. D.3.5

1. A.4
2. B.5
3. C.6
4. D.8

1. A.$3\times 1$
2. B.$1\times 1$
3. C.$1\times 3$
4. D.$3\times 3$

What is ${\int }_{-\pi /2}^{\pi /2}|\sin x|dx$ equal to ?

1. A.2
2. B.1
3. C.$\pi$
4. D.0