MSA Education BD

Integration Quiz

1. The integral of \( \int x^2 dx \) is:

A) \( \frac{1}{3}x^3 + C \) B) \( x^3 + C \) C) \( \frac{1}{2}x^2 + C \) D) \( \frac{1}{4}x^4 + C \)

2. The integral of \( \int \sin x \, dx \) is:

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

3. \( \int e^x \, dx \) equals:

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

4. The integral \( \int \frac{1}{x} \, dx \) is:

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

5. If \( \frac{d}{dx} F(x) = f(x) \), then \( \int f(x) \, dx \) is:

A) \( F(x) + C \) B) \( F(x) \) C) \( F(x) + C \) D) \( F(x) \)

6. The method of integration by parts is based on:

A) Chain rule B) Product rule C) Quotient rule D) Addition rule

7. In the integration by parts formula, \( \int u \, dv = \):

A) \( uv \, \int v \, du \) B) \( uv + \int v \, du \) C) \( vu \, \int u \, dv \) D) \( uv \, v \, du \)

8. The integral \( \int \cos^2(x) \, dx \) can be solved using:

A) Substitution method B) Partial fractions C) Trigonometric identities D) Integration by parts

9. What is the integral of \( \int \frac{1}{x^2 + a^2} \, dx \)?

A) \( \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \) B) \( \frac{1}{a} \sin^{-1}\left(\frac{x}{a}\right) + C \) C) \( \frac{1}{a} \ln\left(x + a\right) + C \) D) \( \frac{1}{a} \cos^{-1}\left(\frac{x}{a}\right) + C \)

10. The integral \( \int e^{2x} \, dx \) is:

A) \( \frac{1}{2} e^{2x} + C \) B) \( 2e^{2x} + C \) C) \( \frac{1}{e^{2x}} + C \) D) \( e^{2x} + C \)

11. The definite integral \( \int_0^1 x^2 \, dx \) equals:

A) \( \frac{1}{3} \) B) \( \frac{1}{2} \) C) \( 1 \) D) \( 0 \)

12. If \( \int_a^b f(x) \, dx = F(b) - F(a) \), then \( F(x) \) is:

A) The derivative of \( f(x) \) B) A constant C) The antiderivative of \( f(x) \) D) The inverse of \( f(x) \)

13. The definite integral \( \int_{-\pi}^{\pi} \sin(x) \, dx \) is:

A) \( 0 \) B) \( 2 \) C) \( -2 \) D) \( \pi \)

14. The area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is given by:

A) \( \int_0^2 x^2 \, dx \) B) \( \int_0^2 2x \, dx \) C) \( \int_0^2 x^3 \, dx \) D) \( \int_0^2 x^4 \, dx \)

15. The value of the definite integral \( \int_0^a f(x) \, dx + \int_a^b f(x) \, dx \) equals:

A) \( \int_0^b f(x) \, dx \) B) \( \int_a^0 f(x) \, dx \) C) \( \int_b^a f(x) \, dx \) D) \( \int_0^b f(x) \, dx + \int_a^0 f(x) \, dx \)

16. The integral \( \int \sqrt{1 - x^2} \, dx \) represents the area of:

A) A circle B) A semicircle C) A parabola D) A hyperbola

17. The integral \( \int_0^\pi \sin^2(x) \, dx \) represents:

A) The average value of \( \sin(x) \) on \( [0, \pi] \) B) The total energy in a sinusoidal wave C) The area under \( \sin^2(x) \) from \( 0 \) to \( \pi \) D) The length of the arc of \( \sin(x) \) from \( 0 \) to \( \pi \)

18. To find the volume of a solid of revolution generated by rotating \( y = f(x) \) about the x-axis, we use:

A) \( \int 2\pi f(x) \, dx \) B) \( \int \pi (f(x))^2 \, dx \) C) \( \int 2\pi x f(x) \, dx \) D) \( \int \pi f(x) \, dx \)

19. The centroid of a region bounded by \( y = f(x) \) and the x-axis can be found using:

A) \( \frac{1}{A} \int x f(x) \, dx \) B) \( \frac{1}{A} \int y f(y) \, dy \) C) \( \frac{1}{A} \int x^2 f(x) \, dx \)

20. The integral \( \int \frac{dx}{\sqrt{a^2 - x^2}} \) is:

A) \( \ln\left|\frac{a + x}{a - x}\right| + C \) B) \( \sin^{-1}\left(\frac{x}{a}\right) + C \) C) \( \cos^{-1}\left(\frac{x}{a}\right) + C \) D) \( \frac{1}{a} \ln\left|\frac{a + x}{a - x}\right| + C \

21. The integral \( \int \frac{1}{\sqrt{x^2 + a^2}} \, dx \) is:

A) \( \ln\left|x + \sqrt{x^2 + a^2}\right| + C \) B) \( \ln\left|x - \sqrt{x^2 + a^2}\right| + C \) C) \( \ln\left|x + a\right| + C \) D) \( \ln\left|x - a\right| + C \)

22. The integral \( \int e^{x} \, dx \) is:

A) \( e^{x} + C \) B) \( e^{x} + C \) C) \( e^x + C \) D) \( e^x + C \)

23. The integral of \( \int \frac{dx}{x \ln(x)} \) is:

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

24. The integral \( \int x \cos(x) \, dx \) can be solved using:

A) Substitution method B) Integration by parts C) Partial fractions D) Trigonometric substitution

25. The area between two curves \( y = f(x) \) and \( y = g(x) \) from \( x = a \) to \( x = b \) is:

A) \( \int_a^b [f(x) - g(x)] \, dx \) B) \( \int_a^b [g(x) - f(x)] \, dx \) C) \( \int_a^b [f(x) + g(x)] \, dx \) D) \( \int_a^b f(x) \, dx + \int_a^b g(x) \, dx \)

26. To find the volume of a solid obtained by rotating a region around the y-axis, the formula used is:

A) \( \pi \int_a^b [f(x)]^2 \, dx \) B) \( \pi \int_a^b [x f(x)]^2 \, dx \) C) \( \pi \int_a^b [f(y)]^2 \, dy \) D) \( \pi \int_a^b x^2 f(x) \, dx \)

27. The integral \( \int \frac{1}{x^2 - a^2} \, dx \) is:

A) \( \frac{1}{2a} \ln\left|\frac{x - a}{x + a}\right| + C \) B) \( \frac{1}{2a} \ln\left|\frac{x + a}{x - a}\right| + C \) C) \( \frac{1}{a} \ln\left|\frac{x + a}{x - a}\right| + C \) D) \( \frac{1}{a} \ln\left|\frac{x - a}{x + a}\right| + C \)

28. The integral \( \int \frac{dx}{x^2 + 2x + 2} \) can be simplified using:

A) Partial fractions B) Trigonometric substitution C) Completing the square D) Integration by parts

29. The integral \( \int \ln(x) \, dx \) is:

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

30. To find the area of a region bounded by \( y = x^2 \) and \( y = x + 2 \), we first:

A) Find the points of intersection B) Integrate \( y = x^2 \) over the given range C) Integrate \( y = x + 2 \) over the given range D) Integrate both curves from \( x = 0 \) to \( x = 2 \)

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