JEE Main - Integral as an antiderivative Online Test (MCQ Quiz)

1. What is the fundamental concept represented by an integral as an antiderivative?

A) Finding the area under a curve B) Reversing the process of differentiation C) Calculating the volume of a solid D) Determining the limit of a function

2. If F(x) is an antiderivative of f(x), what is the relationship between F'(x) and f(x)?

A) F'(x) = f'(x) B) F'(x) = f(x) C) F(x) = f'(x) D) F(x) = F'(x)

3. The notation $\int f(x) dx$ represents:

A) The derivative of f(x) B) The definite integral of f(x) C) The indefinite integral (antiderivative) of f(x) D) The limit of f(x)

4. What is the general form of the antiderivative of a function f(x)?

A) F(x) B) F'(x) C) F(x) + C D) F(x) - C

5. The constant 'C' in the indefinite integral $\int f(x) dx = F(x) + C$ is known as the:

A) Limit of integration B) Constant of proportionality C) Constant of integration D) Variable of integration

6. What is the antiderivative of $x^n$, where $n \neq -1$?

A) $\frac{x^{n+1}}{n+1}$ B) $\frac{x^{n-1}}{n-1}$ C) $nx^{n-1}$ D) $nx^{n+1}$

7. Find the antiderivative of $f(x) = x^2$.

A) $\frac{x^3}{3} + C$ B) $2x + C$ C) $3x^2 + C$ D) $\frac{x^2}{2} + C$

8. What is the antiderivative of $f(x) = \sin(x)$?

A) $\cos(x) + C$ B) $-\cos(x) + C$ C) $\sin(x) + C$ D) $-\sin(x) + C$

9. The antiderivative of $f(x) = \cos(x)$ is:

A) $-\sin(x) + C$ B) $\sin(x) + C$ C) $\cos(x) + C$ D) $-\cos(x) + C$

10. What is the antiderivative of $f(x) = e^x$?

A) $e^x + C$ B) $e^{x+1} + C$ C) $\frac{e^x}{x} + C$ D) $x e^{x-1} + C$

11. Find the antiderivative of $f(x) = \frac{1}{x}$ for $x > 0$.

A) $\ln(x) + C$ B) $\frac{1}{x^2} + C$ C) $x + C$ D) $\ln|x| + C$

12. The antiderivative of $f(x) = \sec^2(x)$ is:

A) $\tan(x) + C$ B) $\cot(x) + C$ C) $\sec(x) + C$ D) $\csc(x) + C$

13. What is the antiderivative of $f(x) = \frac{1}{\sqrt{1-x^2}}$?

A) $\arcsin(x) + C$ B) $\arccos(x) + C$ C) $\arctan(x) + C$ D) $\text{arcsec}(x) + C$

14. If $\int f(x) dx = G(x) + C$, then what is $\int g(x) dx$ if $g(x) = k \cdot f(x)$ for a constant $k$?

A) $k \cdot G(x)$ B) $k \cdot G(x) + C$ C) $\frac{G(x)}{k} + C$ D) $G(x) + kC$

15. What is the property of linearity for indefinite integrals?

A) $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$ B) $\int c \cdot f(x) dx = c \int f(x) dx$ C) Both A and B D) None of the above

16. Find the antiderivative of $f(x) = 3x^4 - 2x + 5$.

A) $12x^3 - 2 + C$ B) $\frac{3x^5}{5} - x^2 + 5x + C$ C) $x^5 - x^2 + 5x + C$ D) $\frac{x^5}{5} - x^2 + 5x + C$

17. The antiderivative of $f(x) = a^x$ (where $a > 0, a \neq 1$) is:

A) $\frac{a^x}{\ln(a)} + C$ B) $a^x \ln(a) + C$ C) $\frac{a^{x+1}}{x+1} + C$ D) $a^x + C$

18. What is the antiderivative of $f(x) = \frac{1}{1+x^2}$?

A) $\arctan(x) + C$ B) $\text{arccot}(x) + C$ C) $\arcsin(x) + C$ D) $\arccos(x) + C$

19. Find the antiderivative of $f(x) = \sinh(x)$.

A) $\cosh(x) + C$ B) $-\cosh(x) + C$ C) $\sinh(x) + C$ D) $e^x + C$

20. What is the antiderivative of $f(x) = \cosh(x)$?

A) $\sinh(x) + C$ B) $-\sinh(x) + C$ C) $\cosh(x) + C$ D) $e^{-x} + C$

21. If $F(x)$ is an antiderivative of $f(x)$, what is $F(x) + C$?

A) A specific antiderivative B) The general antiderivative C) The derivative of f(x) D) The definite integral of f(x)

22. The antiderivative of $f(x) = \frac{1}{x}$ for $x < 0$ is:

A) $\ln(-x) + C$ B) $\ln(x) + C$ C) $\ln|x| + C$ D) $\frac{1}{x^2} + C$

23. What is the antiderivative of $f(x) = x^{-1}$?

A) $\ln|x| + C$ B) $\frac{x^0}{0} + C$ C) $\ln(x) + C$ D) $x + C$

24. Find the antiderivative of $f(x) = \sqrt{x}$.

A) $\frac{2}{3} x^{3/2} + C$ B) $\frac{3}{2} x^{3/2} + C$ C) $2x^{1/2} + C$ D) $\frac{1}{2} x^{-1/2} + C$

25. The antiderivative of $f(x) = \frac{1}{\sqrt{x}}$ is:

A) $2\sqrt{x} + C$ B) $\frac{1}{2\sqrt{x}} + C$ C) $\sqrt{x} + C$ D) $\frac{1}{3}x^{3/2} + C$

26. What is the antiderivative of $f(x) = x^3$?

A) $3x^2 + C$ B) $\frac{x^4}{4} + C$ C) $x^4 + C$ D) $\frac{x^3}{3} + C$

27. Find the antiderivative of $f(x) = \sin(2x)$.

A) $-\frac{1}{2}\cos(2x) + C$ B) $\frac{1}{2}\cos(2x) + C$ C) $-2\cos(2x) + C$ D) $2\cos(2x) + C$

28. The antiderivative of $f(x) = e^{3x}$ is:

A) $\frac{1}{3}e^{3x} + C$ B) $3e^{3x} + C$ C) $e^{3x} + C$ D) $\frac{e^{3x}}{3} + C$

29. What is the antiderivative of $f(x) = \tan(x)$?

30. Find the antiderivative of $f(x) = \cot(x)$.

31. The antiderivative of $f(x) = \sec(x)$ is:

32. What is the antiderivative of $f(x) = \csc(x)$?

33. If $\int f(x) dx = F(x) + C$, then what is $\int (f(x) + g(x)) dx$?

A) $F(x) + \int g(x) dx$ B) $F(x) + g(x) + C$ C) $F(x) + G(x) + C$ D) $F(x) + C + G(x) + C$

34. What is the antiderivative of $f(x) = \frac{1}{x^2}$?

A) $-\frac{1}{x} + C$ B) $\frac{1}{x} + C$ C) $\frac{1}{x^3} + C$ D) $-\frac{1}{x^3} + C$

35. Find the antiderivative of $f(x) = \frac{1}{x^3}$.

A) $-\frac{1}{2x^2} + C$ B) $\frac{1}{2x^2} + C$ C) $\frac{1}{x^4} + C$ D) $-\frac{1}{x^4} + C$

36. The antiderivative of $f(x) = x^n$ for $n=-1$ is:

A) $\ln|x| + C$ B) $\frac{x^0}{0} + C$ C) $x + C$ D) $\frac{x^{-2}}{-2} + C$

37. What is the antiderivative of $f(x) = c$, where c is a constant?

A) $cx + C$ B) $c + C$ C) $x + C$ D) $0 + C$

38. If $F'(x) = f(x)$, then the indefinite integral of $f(x)$ is:

A) $F'(x) + C$ B) $F''(x) + C$ C) $F(x) + C$ D) $f(x) + C$

39. The antiderivative of $f(x) = \frac{1}{\sqrt{1-x^2}}$ is also known as:

A) $\arccos(x) + C$ B) $\arctan(x) + C$ C) $-\arccos(x) + C$ D) $\text{arcsec}(x) + C$

40. What is the antiderivative of $f(x) = \frac{x}{\sqrt{1-x^2}}$?

A) $-\sqrt{1-x^2} + C$ B) $\sqrt{1-x^2} + C$ C) $\frac{1}{2\sqrt{1-x^2}} + C$ D) $2\sqrt{1-x^2} + C$

41. Find the antiderivative of $f(x) = \frac{1}{x \ln(x)}$.

A) $\ln|\ln(x)| + C$ B) $\ln(x) + C$ C) $\frac{1}{\ln(x)} + C$ D) $\ln|x| + C$

42. The antiderivative of $f(x) = \frac{2x}{1+x^2}$ is:

A) $\ln(1+x^2) + C$ B) $\ln|1+x^2| + C$ C) $\frac{1}{1+x^2} + C$ D) $\arctan(x^2) + C$

43. What is the antiderivative of $f(x) = \frac{1}{x \sqrt{x^2-1}}$?

A) $\text{arcsec}(|x|) + C$ B) $\text{arccsc}(|x|) + C$ C) $\ln|x| + C$ D) $\frac{1}{x} + C$

44. Find the antiderivative of $f(x) = \frac{1}{x^2+a^2}$ where $a \neq 0$.

A) $\frac{1}{a}\arctan(\frac{x}{a}) + C$ B) $\arctan(\frac{x}{a}) + C$ C) $\frac{1}{a}\arcsin(\frac{x}{a}) + C$ D) $\frac{1}{a}\text{arcsec}(\frac{x}{a}) + C$

45. The antiderivative of $f(x) = \frac{1}{\sqrt{a^2-x^2}}$ where $a > 0$ is:

A) $\arcsin(\frac{x}{a}) + C$ B) $\frac{1}{a}\arcsin(\frac{x}{a}) + C$ C) $\arccos(\frac{x}{a}) + C$ D) $\frac{1}{a}\arccos(\frac{x}{a}) + C$

46. What is the antiderivative of $f(x) = \frac{1}{x^2-a^2}$?

A) $\frac{1}{2a}\ln|\frac{x-a}{x+a}| + C$ B) $\frac{1}{2a}\ln|\frac{x+a}{x-a}| + C$ C) $\frac{1}{a}\ln|x^2-a^2| + C$ D) $\frac{1}{a}\arctan(\frac{x}{a}) + C$

47. If $\int f(x) dx = F(x) + C$, then $\int f(ax+b) dx$ is:

A) $\frac{1}{a}F(ax+b) + C$ B) $aF(ax+b) + C$ C) $F(ax+b) + C$ D) $\frac{1}{a}F(x) + C$

48. Consider the function $f(x) = \frac{d}{dx}(\sin(x^2))$. What is the integral of $f(x)$?

A) $\sin(x^2) + C$ B) $\cos(x^2) + C$ C) $2x\cos(x^2) + C$ D) $\frac{1}{2x}\sin(x^2) + C$

49. The antiderivative of $f(x) = x \cos(x)$ is found using integration by parts. The result is:

A) $x \sin(x) + \cos(x) + C$ B) $x \sin(x) - \cos(x) + C$ C) $\sin(x) + x \cos(x) + C$ D) $\sin(x) - x \cos(x) + C$

50. What is the antiderivative of $f(x) = \ln(x)$?

A) $x \ln(x) - x + C$ B) $x \ln(x) + x + C$ C) $\frac{1}{x} + C$ D) $\ln(x) + C$

Integral as an antiderivative - Online Test

Test Results