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HPCET Integrals — practice questions — page 2

Explore 56 free Integrals questions in HPCET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Integrals questions and exercises with answers

Showing questions 31–56 of 56.

By partial fractions, ∫ dx/(x(x+1)) equals:Read the answer and worked solution →∫₀^(2π) cos x dx equals:Read the answer and worked solution →Area under y = x² from x = 0 to x = 2:Read the answer and worked solution →∫ x sin x dx (by parts):Read the answer and worked solution →∫ dx/(a² - x²) (for |x| < a):Read the answer and worked solution →∫₀^(π/2) sin x cos x dx equals:Read the answer and worked solution →Complete the relationship between the two calculiRead the answer and worked solution →True or false: the constant of integrationRead the answer and worked solution →Differentiation or integration?Read the answer and worked solution →Tap every property of indefinite integrals the chapter givesRead the answer and worked solution →Properties in useRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Order the steps: check an indefinite integralRead the answer and worked solution →Flip and pair: term with meaningRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →True or false: reading the notationRead the answer and worked solution →Definite or indefinite?Read the answer and worked solution →Reading the definitions backRead the answer and worked solution →Complete the linearity rulesRead the answer and worked solution →Which process is being described?Read the answer and worked solution →True or false: what the constant meansRead the answer and worked solution →Checking an integralRead the answer and worked solution →Tap every true statement about integrationRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →

Worked practice questions

By partial fractions, ∫ dx/(x(x+1)) equals:

  1. ln |x| + C
  2. ln |x(x+1)| + C
  3. ln |x/(x+1)| + C
  4. arctan x + C
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Correct answer: C. ln |x/(x+1)| + C

1/(x(x+1)) = 1/x - 1/(x+1). We have ∫ = ln |x| - ln |x+1| + C. So ∫ = ln |x/(x+1)| + C.

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∫₀^(2π) cos x dx equals:

  1. -2π
  2. 2
  3. 0
  4. 2π
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Correct answer: C. 0

∫₀^(2π) cos x dx = [sin x]₀^(2π) = sin(2π) - sin(0) = 0 - 0 = 0. Cosine integrated over a full period equals zero.

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Area under y = x² from x = 0 to x = 2:

  1. 4
  2. 2
  3. 4/3
  4. 8/3
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Correct answer: D. 8/3

∫₀² x² dx = [x³/3]₀² = 8/3.

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