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Integrals

Each subtopic includes About section, revision page link, 10 preview questions, and practice CTAs.

Integration as inverse process of differentiation

Subtopic

Integration as inverse process of differentiation under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the anti-derivative of 8x78x^7.

    A.

    x8+Cx^8 + C

    B.

    56x6+C56x^6 + C

    C.

    8x8+C8x^8 + C

    D.

    x88+C\frac{x^8}{8} + C

  2. 2.

    Find the anti-derivative of 11+x2\frac{1}{1+x^2}.

    A.

    sin⁡−1x+C\sin^{-1} x + C

    B.

    log⁡(1+x2)+C\log(1+x^2) + C

    C.

    tan⁡−1x+C\tan^{-1} x + C

    D.

    cot⁡−1x+C\cot^{-1} x + C

  3. 3.

    Find the primitive of 11−x2\frac{1}{\sqrt{1-x^2}} for ∣x∣<1|x| < 1.

    A.

    cos⁡−1x+C\cos^{-1} x + C

    B.

    sin⁡−1x+C\sin^{-1} x + C

    C.

    tan⁡−1x+C\tan^{-1} x + C

    D.

    sec⁡−1x+C\sec^{-1} x + C

Download the worksheet for Integrals - Integration as inverse process of differentiation to practice offline. It includes additional chapter-level practice questions.

Integration of a variety of functions by substitution, by partial fractions and by parts

Subtopic

Integration of a variety of functions by substitution, by partial fractions and by parts under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the integral ∫1x2−6x+10dx\int \frac{1}{x^2-6x+10} dx.

    A.

    log⁡∣x−3∣+C\log|x-3| + C

    B.

    12log⁡∣x2−6x+10∣+C\frac{1}{2} \log|x^2-6x+10| + C

    C.

    tan⁡−1(x−3)+C\tan^{-1}(x-3) + C

    D.

    tan⁡−1(x+3)+C\tan^{-1}(x+3) + C

  2. 2.

    Evaluate ∫1x2+4x+5dx\int \frac{1}{x^2+4x+5} dx by completing the square.

    A.

    tan⁡−1(x+2)+C\tan^{-1}(x+2) + C

    B.

    log⁡∣x2+4x+5∣+C\log|x^2+4x+5| + C

    C.

    12tan⁡−1(x+2)+C\frac{1}{2} \tan^{-1}(x+2) + C

    D.

    sin⁡−1(x+2)+C\sin^{-1}(x+2) + C

  3. 3.

    Evaluate ∫xsec⁡2xdx\int x \sec^2 x dx using integration by parts.

    A.

    xtan⁡x+log⁡∣cos⁡x∣+Cx \tan x + \log|\cos x| + C

    B.

    xtan⁡x−log⁡∣cos⁡x∣+Cx \tan x - \log|\cos x| + C

    C.

    xsec⁡x−tan⁡x+Cx \sec x - \tan x + C

    D.

    tan⁡x+xlog⁡∣sec⁡x∣+C\tan x + x \log|\sec x| + C

Download the worksheet for Integrals - Integration of a variety of functions by substitution, by partial fractions and by parts to practice offline. It includes additional chapter-level practice questions.

Evaluation of simple integrals and problems based on them

Subtopic

Evaluation of simple integrals and problems based on them under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Evaluate: ∫(ex−sin⁡x)dx\int (e^x - \sin x) dx.

    A.

    ex+cos⁡x+Ce^x + \cos x + C

    B.

    ex−cos⁡x+Ce^x - \cos x + C

    C.

    ex+sin⁡x+Ce^x + \sin x + C

    D.

    ex−sin⁡x+Ce^x - \sin x + C

  2. 2.

    Evaluate the integral: ∫13xdx\int \frac{1}{3x} dx.

    A.

    13ln⁡∣x∣+C\frac{1}{3} \ln |x| + C

    B.

    3ln⁡∣x∣+C3 \ln |x| + C

    C.

    ln⁡∣3x∣+C\ln |3x| + C

    D.

    13x2+C\frac{1}{3x^2} + C

  3. 3.

    Find the integral: ∫(x3−8)dx\int (x^3 - 8) dx.

    A.

    x44−8x+C\frac{x^4}{4} - 8x + C

    B.

    3x2+C3x^2 + C

    C.

    x44+C\frac{x^4}{4} + C

    D.

    x4−8x+Cx^4 - 8x + C

Download the worksheet for Integrals - Evaluation of simple integrals and problems based on them to practice offline. It includes additional chapter-level practice questions.

Definite integrals as a limit of a sum

Subtopic

Definite integrals as a limit of a sum under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    The value of the limit lim⁡n→∞1n∑r=1n(rn)8\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} (\frac{r}{n})^8 is:

    A.

    18\frac{1}{8}

    B.

    19\frac{1}{9}

    C.

    110\frac{1}{10}

    D.

    17\frac{1}{7}

  2. 2.

    Evaluate lim⁡n→∞1n∑r=1ncos⁡(rπn)\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} \cos\left(\frac{r\pi}{n}\right).

    A.

    11

    B.

    00

    C.

    π\pi

    D.

    1π\frac{1}{\pi}

  3. 3.

    In the limit of a sum, if h=0.1/nh = 0.1/n, what is the length of the interval (b−a)(b-a)?

    A.

    11

    B.

    0.10.1

    C.

    1010

    D.

    nn

Download the worksheet for Integrals - Definite integrals as a limit of a sum to practice offline. It includes additional chapter-level practice questions.

Fundamental Theorem of Calculus

Subtopic

Fundamental Theorem of Calculus under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the value of ∫123x4 dx\int_1^2 \frac{3}{x^4} \, dx.

    A.

    78\frac{7}{8}

    B.

    1516\frac{15}{16}

    C.

    12\frac{1}{2}

    D.

    34\frac{3}{4}

  2. 2.

    Calculate ∫0π/2sin⁡x dx\int_0^{\pi/2} \sin x \, dx.

    A.

    00

    B.

    11

    C.

    −1-1

    D.

    π\pi

  3. 3.

    Evaluate ∫01(x2+x+1) dx\int_0^1 (x^2 + x + 1) \, dx.

    A.

    116\frac{11}{6}

    B.

    56\frac{5}{6}

    C.

    76\frac{7}{6}

    D.

    22

Download the worksheet for Integrals - Fundamental Theorem of Calculus to practice offline. It includes additional chapter-level practice questions.

Basic properties of definite integrals and evaluation of definite integrals

Subtopic

Basic properties of definite integrals and evaluation of definite integrals under Integrals for Grade 12 CBSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    According to the properties of definite integrals, ∫abf(x)dx\int_{a}^{b} f(x) dx is equal to:

    A.

    ∫abf(a+b−x)dx\int_{a}^{b} f(a+b-x) dx

    B.

    ∫abf(x−a−b)dx\int_{a}^{b} f(x-a-b) dx

    C.

    ∫abf(a−x)dx\int_{a}^{b} f(a-x) dx

    D.

    ∫abf(b−x)dx\int_{a}^{b} f(b-x) dx

  2. 2.

    Evaluate: ∫01(x+1)2dx\int_{0}^{1} (x + 1)^2 dx.

    A.

    73\frac{7}{3}

    B.

    83\frac{8}{3}

    C.

    1

    D.

    2

  3. 3.

    Find the value of ∫141xdx\int_{1}^{4} \frac{1}{\sqrt{x}} dx.

    A.

    1

    B.

    2

    C.

    3

    D.

    4

Download the worksheet for Integrals - Basic properties of definite integrals and evaluation of definite integrals to practice offline. It includes additional chapter-level practice questions.

Integrals - CBSE Class 12 Maths Notes & Revision | Krit.club