Sure, here are 50 multiple-choice questions on integration:
1. What is the integral of \( \int x^2 \, dx \)?
a) \( \frac{x^3}{3} + C \)
b) \( \frac{x^2}{2} + C \)
c) \( x^3 + C \)
d) \( 2x + C \)
2. The integral of \( \int e^x \, dx \) is:
a) \( e^x + C \)
b) \( e^{2x} + C \)
c) \( \frac{e^x}{x} + C \)
d) \( \ln(e^x) + C \)
3. What is the integral of \( \int \cos x \, dx \)?
a) \( \sin x + C \)
b) \( -\sin x + C \)
c) \( -\cos x + C \)
d) \( \cos x + C \)
4. The integral of \( \int \frac{1}{x} \, dx \) is:
a) \( \ln|x| + C \)
b) \( \frac{1}{x} + C \)
c) \( e^x + C \)
d) \( x \ln x + C \)
5. Find the integral of \( \int 2x \, dx \):
a) \( x^2 + C \)
b) \( 2x^2 + C \)
c) \( x + C \)
d) \( x^2 + 2C \)
6. What is the integral of \( \int \sin x \, dx \)?
a) \( \cos x + C \)
b) \( -\cos x + C \)
c) \( \sin x + C \)
d) \( -\sin x + C \)
7. The integral of \( \int x e^x \, dx \) can be solved using:
a) Substitution
b) Integration by parts
c) Partial fractions
d) Trigonometric substitution
8. Find the integral of \( \int \frac{x^2}{x+1} \, dx \):
a) \( \frac{x^2}{2} + \ln|x+1| + C \)
b) \( \frac{x^3}{3} - \ln|x+1| + C \)
c) \( x^2 - \ln|x+1| + C \)
d) \( \frac{x^2}{2} + x - \ln|x+1| + C \)
9. The integral of \( \int \tan x \, dx \) is:
a) \( -\ln|\cos x| + C \)
b) \( \ln|\sin x| + C \)
c) \( \ln|\cos x| + C \)
d) \( \ln|\sec x| + C \)
10. What is the integral of \( \int x e^{-x} \, dx \)?
a) \( -x e^{-x} - e^{-x} + C \)
b) \( x e^{-x} + e^{-x} + C \)
c) \( -x e^{-x} + e^{-x} + C \)
d) \( x e^{-x} - e^{-x} + C \)
11. Find the integral of \( \int \sec^2 x \, dx \):
a) \( \tan x + C \)
b) \( \sec x + C \)
c) \( \sin x + C \)
d) \( -\tan x + C \)
12. The integral of \( \int x \ln x \, dx \) can be solved using:
a) Integration by parts
b) Substitution
c) Partial fractions
d) Trigonometric substitution
13. What is the integral of \( \int \frac{1}{\sqrt{x}} \, dx \)?
a) \( 2\sqrt{x} + C \)
b) \( \sqrt{x} + C \)
c) \( -2\sqrt{x} + C \)
d) \( \frac{1}{2} \sqrt{x} + C \)
14. The integral of \( \int \frac{x}{x^2 + 1} \, dx \) is:
a) \( \frac{1}{2} \ln(x^2 + 1) + C \)
b) \( \ln|x^2 + 1| + C \)
c) \( \arctan x + C \)
d) \( \frac{1}{x^2 + 1} + C \)
15. Find the integral of \( \int \frac{e^x}{e^x + 1} \, dx \):
a) \( \ln(e^x + 1) + C \)
b) \( x - \ln(e^x + 1) + C \)
c) \( \ln|e^x - 1| + C \)
d) \( \ln|e^x + 1| - x + C \)
16. The integral of \( \int \frac{dx}{x^2 - a^2} \) 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) \( \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C \)
d) \( \frac{1}{2a} \arctan \left( \frac{x}{a} \right) + C \)
17. What is the integral of \( \int \frac{1}{x^2 + a^2} \, dx \)?
a) \( \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C \)
b) \( \frac{1}{x} \arctan \left( \frac{x}{a} \right) + C \)
c) \( \frac{1}{a} \ln \left| x^2 + a^2 \right| + C \)
d) \( \arctan \left( \frac{x}{a} \right) + C \)
18. The integral of \( \int \frac{x^2}{x^2 + 1} \, dx \) is:
a) \( x - \ln|x^2 + 1| + C \)
b) \( x + \ln|x^2 + 1| + C \)
c) \( \frac{x^3}{3} - \frac{1}{2} \ln|x^2 + 1| + C \)
d) \( \frac{x^2}{2} + \ln|x^2 + 1| + C \)
19. What is the integral of \( \int \frac{e^x}{e^{2x} + 1} \, dx \)?
a) \( \frac{1}{2} \ln(e^{2x} + 1) + C \)
b) \( \frac{1}{2} \arctan(e^x) + C \)
c) \( \arctan(e^x) + C \)
d) \( \ln|e^x + \sqrt{e^{2x} + 1}| + C \)
20. Find the integral of \( \int \frac{dx}{\sqrt{x^2 + a^2}} \):
a) \( \ln \left| x + \sqrt{x^2 + a^2} \right| + C \)
b) \( \ln \left| x - \sqrt{x^2 + a^2} \right| + C \)
c) \( \arcsinh \left( \frac{x}{a} \right) + C \)
d) \( \frac{1}{a} \ln \left| x + \sqrt{x^2 + a^2} \right| + C \)
21. The integral of \( \int \ln x \, dx \) is:
a) \( x \ln x - x + C \)
b) \( \frac{x^2}{2} \ln x + C \)
c) \( \frac{x \ln x - x}{2} + C \)
d) \( x \ln x + C \)
22. What is the integral of \( \int x \sin x \, dx \)?
a) \( -x \cos x + \sin x + C \)
b) \( x \cos x - \sin x + C \)
c) \( -x \cos x - \sin x + C \)
d) \( x \sin x + \cos x + C \)
23. The integral of \( \int e^{-x^2} \, dx \) is:
a) \( \frac{\sqrt{\pi}}{2} \
