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ISC Class 12 Integrals and Applications — practice questions — page 2

Explore 97 free Integrals and Applications questions in ISC Class 12 Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Integrals and Applications questions and exercises with answers

Showing questions 31–60 of 97.

Order the area methodRead the answer and worked solution →Order the steps: check an indefinite integralRead the answer and worked solution →Checking an integralRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →By Walli's formula, ∫_0^(π/2) sin^4(x) dx equals:Read the answer and worked solution →Compute ∫ 2x sin(x²) dx using substitution:Read the answer and worked solution →The integral ∫_{-1}^{1} x³ dx equals:Read the answer and worked solution →Complete the linearity rulesRead 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 →Three of these are wrong. Find the one that holds.Read 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 →Reading the definitions backRead the answer and worked solution →True or false: what the constant meansRead the answer and worked solution →Tap every true statement about integrationRead the answer and worked solution →To evaluate $\int x\cos x\, dx$, the best technique is:Read the answer and worked solution →By the Fundamental Theorem of Calculus, ∫_0^2 (3x² + 2x) dx equals:Read the answer and worked solution →Tap every property of indefinite integrals the chapter givesRead the answer and worked solution →The integral $\int x^3\, dx$ equals:Read the answer and worked solution →The AREA bounded by y = x² and the x-axis from x = 0 to x = 3 is:Read the answer and worked solution →The improper integral ∫_1^∞ (1/x²) dx evaluates to:Read the answer and worked solution →$\displaystyle\int x^n\,dx$ (where $n \ne -1$) equals:Read the answer and worked solution →The integral ∫ x^n dx (for n ≠ -1) equals:Read the answer and worked solution →Using integration by parts, $\displaystyle\int x e^x\,dx$ equals:Read the answer and worked solution →To solve the homogeneous differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, the appropriate substitution is:Read the answer and worked solution →Using partial fractions, $\displaystyle\int \dfrac{1}{(x-1)(x-2)}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{x^2 - a^2}\,dx$ (with $|x| > a > 0$) equals:Read the answer and worked solution →

Worked practice questions

Order the area method

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Steps
  1. Sketch the region
  2. Choose a thin strip
  3. Write dA for it
  4. Integrate over limits
  5. Take absolute values
Instructions
Tap the steps in the order they are done.
_Source: syllabus Mathematics, Ch 8 'Application of Integrals'_

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Order the steps: check an indefinite integral

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Steps
  1. Write down your proposed integral, with its constant
  2. Differentiate that expression
  3. The constant differentiates away to zero
  4. Compare the result with the original integrand
  5. If they match, the integral is correct
Differentiate the answer and compare. Source: syllabus Maths, Ch 7 Integrals.

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Checking an integral

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Questions
  1. Question
    The quickest check on an indefinite integral is to
    Choices
    1. integrate again
    2. differentiate the answer
    3. substitute zero
    Correct choice
    differentiate the answer
  2. Question
    On differentiating, the constant becomes
    Choices
    1. one
    2. itself
    3. zero
    Correct choice
    zero
  3. Question
    If the derivative matches the integrand, the answer is
    Choices
    1. correct
    2. incorrect
    3. undecidable
    Correct choice
    correct
Differentiate and compare. Source: syllabus Maths, Ch 7 Integrals.

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