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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