JEE (Main) Mathematics - Integration MCQs (16-30)
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16. The integral \( \int \frac{1}{x \ln(x)} \, dx \) can be evaluated using:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: a) Substitution
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17. The integral \( \int \frac{x^2}{x^3 + 1} \, dx \) can be solved using:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: a) Substitution
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18. Which substitution is used to solve \( \int \frac{1}{\sqrt{1-x^2}} \, dx \)?
- a) \( x = \sin(\theta) \)
- b) \( x = \tan(\theta) \)
- c) \( x = \cos(\theta) \)
- d) \( x = \sec(\theta) \)
Answer: a) \( x = \sin(\theta) \)
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19. The integral \( \int \ln(x) \, dx \) is best solved using:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: b) Integration by parts
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20. The integral \( \int \frac{dx}{x^2 + 1} \) is evaluated as:
- a) \( \tan^{-1}(x) + C \)
- b) \( \ln(x) + C \)
- c) \( \frac{1}{x} + C \)
- d) \( \sin^{-1}(x) + C \)
Answer: a) \( \tan^{-1}(x) + C \)
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21. Which method is used to solve \( \int \frac{1}{x(x-1)} \, dx \)?
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: c) Partial fractions
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22. To evaluate \( \int x e^{x^2} \, dx \), which substitution is used?
- a) \( u = x^2 \)
- b) \( u = e^x \)
- c) \( u = x \)
- d) \( u = \ln(x) \)
Answer: a) \( u = x^2 \)
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23. The integral \( \int \sin^2(x) \, dx \) can be simplified using which identity?
- a) \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \)
- b) \( \sin^2(x) = \frac{1 + \cos(2x)}{2} \)
- c) \( \sin^2(x) = 1 - \cos^2(x) \)
- d) \( \sin^2(x) = \cos^2(x) \)
Answer: a) \( \sin^2(x) = \frac{1 - \cos(2x)}{2} \)
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24. The integral \( \int \frac{dx}{x^2 + a^2} \) is solved using:
- a) Integration by parts
- b) Substitution
- c) Partial fractions
- d) Standard formula
Answer: d) Standard formula
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25. The integral \( \int \frac{1}{(x+1)(x+2)} \, dx \) is best solved by:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: c) Partial fractions
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26. The integral \( \int e^{x} \sin(x) \, dx \) can be solved using:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: b) Integration by parts
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27. The integral \( \int \frac{1}{\sqrt{x^2 + 2x + 5}} \, dx \) is best solved by:
- a) Completing the square
- b) Integration by parts
- c) Substitution
- d) Trigonometric identities
Answer: a) Completing the square
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28. To evaluate \( \int \frac{dx}{x^2 + a^2} \), we use:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Standard result
Answer: d) Standard result
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29. The integral \( \int \sin(x) \cos(x) \, dx \) can be solved by using which identity?
- a) \( \sin(x) \cos(x) = \frac{\sin(2x)}{2} \)
- b) \( \sin(x) \cos(x) = \frac{1 - \cos(2x)}{2} \)
- c) \( \sin(x) \cos(x) = \frac{1 + \cos(2x)}{2} \)
- d) None of the above
Answer: a) \( \sin(x) \cos(x) = \frac{\sin(2x)}{2} \)
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30. The integral \( \int \frac{1}{x(x+1)} \, dx \) is best solved using:
- a) Substitution
- b) Integration by parts
- c) Partial fractions
- d) Trigonometric identities
Answer: c) Partial fractions
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