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CBSE Class 12 (Commerce) Integrals — practice questions — page 3

Explore 102 free Integrals questions in CBSE Class 12 (Commerce) 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 61–90 of 102.

Integrate 1/(x^2 - 4) by the (1/2a) log formula.Read the answer and worked solution →With a positive, 1/(a^2 - x^2) integrates to:Read the answer and worked solution →Completing the square, integrate 1/(x^2 + 2x + 5):Read the answer and worked solution →What is the integral of (2x + 3)/(x^2 + 3x + 7)?Read the answer and worked solution →A ratio P(x)/Q(x) with deg P less than deg Q is called:Read the answer and worked solution →To integrate P/Q when deg P exceeds deg Q, begin by:Read the answer and worked solution →Resolve (x + 3)/((x - 1)(x + 2)): what multiplies 1/(x - 1)?Read the answer and worked solution →Which sum of partial fractions equals 1/(x(x + 1))?Read the answer and worked solution →Integrating the partial fractions of 1/(x(x + 1)) gives:Read the answer and worked solution →Which setup decomposes 1/((x-1)^2 (x+2))?Read the answer and worked solution →Choose the correct decomposition of 1/((x-1)(x^2+1)).Read the answer and worked solution →Integration by parts says the integral of f g equals:Read the answer and worked solution →Find the integral of x e^x by taking x as the first function.Read the answer and worked solution →Parts on x cos x: what is its antiderivative?Read the answer and worked solution →What do we get on integrating x sin x by parts?Read the answer and worked solution →Integrating log x by parts, the usual choice is:Read the answer and worked solution →Compute the antiderivative of log x for x > 0.Read the answer and worked solution →Parts on ln x times x^2: which function should go first?Read the answer and worked solution →Integrate tan^-1 x using 1 as the second function:Read the answer and worked solution →Find the integral of x^2 e^x by repeating parts twice.Read the answer and worked solution →For e^x [f(x) + f'(x)], the integral is:Read the answer and worked solution →Integrate e^x (tan x + sec^2 x).Read the answer and worked solution →Integrate e^x (1/x - 1/x^2) over x > 0.Read the answer and worked solution →In a definite integral from a to b, the numbers a and b are the:Read the answer and worked solution →Geometrically, the integral of f from a to b measures:Read the answer and worked solution →If A(x) is the area function of continuous f, then A'(x) =Read the answer and worked solution →Find the area under y = x^2 between x = 0 and x = 1.Read the answer and worked solution →Integral of 3x^2 from x = 0 to x = 2 equals:Read the answer and worked solution →What does the integral of sin x give between 0 and pi?Read the answer and worked solution →Value of the integral from 0 to pi/2 of cos x:Read the answer and worked solution →

Worked practice questions

Integrate 1/(x^2 - 4) by the (1/2a) log formula.

  1. (1/2) log |(x-2)/(x+2)|
  2. (1/4) log |(x-2)/(x+2)|
  3. (1/4) tan^-1 (x/2)
  4. (1/2) log |(x+2)/(x-2)|
Show answer and solution

Correct answer: B. (1/4) log |(x-2)/(x+2)|

Using 1/(x^2 - a^2) with a = 2 gives (1/2a) log |(x - a)/(x + a)| = (1/4) log |(x - 2)/(x + 2)|.

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With a positive, 1/(a^2 - x^2) integrates to:

  1. (1/a) sin^-1 (x/a)
  2. (1/2a) tan^-1 (x/a)
  3. (1/2a) log |(a+x)/(a-x)|
  4. (1/2a) log |(x-a)/(x+a)|
Show answer and solution

Correct answer: C. (1/2a) log |(a+x)/(a-x)|

Splitting 1/(a^2 - x^2) into partial fractions gives (1/2a) log |(a + x)/(a - x)|.

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Completing the square, integrate 1/(x^2 + 2x + 5):

  1. (1/2) log (x^2+2x+5)
  2. (1/2) tan^-1 (x/2)
  3. (1/2) tan^-1 ((x+1)/2)
  4. tan^-1 ((x + 1)/2)
Show answer and solution

Correct answer: C. (1/2) tan^-1 ((x+1)/2)

Completing the square gives (x + 1)^2 + 4, so the integral is (1/2) tan^-1 ((x + 1)/2).

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