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JEE Main Integrals and Applications — practice questions — page 4

Explore 98 free Integrals and Applications questions in JEE Main 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 91–98 of 98.

$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:Read the answer and worked solution →Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:Read the answer and worked solution →Properties in useRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Which process is being described?Read the answer and worked solution →The number of arbitrary constants in the general solution of a differential equation of order $n$ is:Read the answer and worked solution →True or false: calculusRead the answer and worked solution →The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$ is:Read the answer and worked solution →

Worked practice questions

$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:

  1. $f(x) f'(x) + C$
  2. $e^x f(x) + C$
  3. $e^x f'(x) + C$
  4. $e^x\bigl(f(x) - f'(x)\bigr) + C$
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Correct answer: B. $e^x f(x) + C$

This is the standard 'product with exponential' identity: $\dfrac{d}{dx}\bigl(e^x f(x)\bigr) = e^x f(x) + e^x f'(x) = e^x(f(x) + f'(x))$ so by anti-differentiation the integral is $e^x f(x) + C$. A useful trick whenever an integrand splits as $e^x$ times (something + its derivative).

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Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:

  1. $\dfrac{1}{2}\log(1 + x^2) + C$
  2. $\dfrac{1}{2}\tan^{-1}(x^2) + C$
  3. $\log(1 + x^2) + C$
  4. $\tan^{-1} x + C$
Show answer and solution

Correct answer: A. $\dfrac{1}{2}\log(1 + x^2) + C$

With $t = 1 + x^2$, $dt = 2x\,dx$, so $\int \dfrac{x}{1+x^2}\,dx = \dfrac{1}{2}\int \dfrac{dt}{t} = \dfrac{1}{2}\log|t| + C = \dfrac{1}{2}\log(1 + x^2) + C$. Note the $1/2$ factor, which trips up answers like option B.

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Properties in use

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Questions
  1. Question
    The integral of f(x) + g(x) equals
    Choices
    1. the sum of their integrals
    2. the product of their integrals
    3. neither
    Correct choice
    the sum of their integrals
  2. Question
    For a real number k, the integral of k·f(x) equals
    Choices
    1. the integral of f
    2. k times the integral of f
    3. k
    Correct choice
    k times the integral of f
  3. Question
    Integration is the inverse of
    Choices
    1. multiplication
    2. substitution
    3. differentiation
    Correct choice
    differentiation
  4. Question
    The C in an indefinite integral is the
    Choices
    1. constant of integration
    2. coefficient
    3. curve
    Correct choice
    constant of integration
The two rules the chapter lists first. Source: syllabus Maths, Ch 7 Integrals.

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