Mathematics

Evaluate the following integrals
$$\int { \cfrac { 1 }{ p+q\tan { x }  }  } dx\quad $$


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Subjective Medium Published on 17th 09, 2020
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Q1 Single Correct Medium
Let $$f\left( x \right) =\frac { \sin { x }  }{ x }$$, then $$\int _{ 0 }^{ \frac { \pi  }{ 2 }  }{ f\left( x \right) f\left( \frac { \pi  }{ 2 } -x \right)  } dx=$$
  • A. $$\int _{ 0 }^{ \pi }{ f\left( x \right) dx }$$
  • B. $$\pi \int _{ 0 }^{ \pi }{ f\left( x \right) dx }$$
  • C. $$\frac { { 1 } }{ \pi } \int _{ 0 }^{ \pi }{ f\left( x \right) dx }$$
  • D. $$\frac { { 2 } }{ \pi } \int _{ 0 }^{ \pi }{ f\left( x \right)dx }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q2 Subjective Medium
Integrate the following w.r.t.$$\displaystyle \int x^2\left(1-\dfrac{2}{x}\right)^2dx $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q3 Assertion & Reason Hard
ASSERTION

$$\displaystyle \int_{-1}^{1}\frac{\sin x-x^{4}}{4-\left | x \right |}dx$$ is same as $$\displaystyle \int_{0}^{1}\frac{-2x^{4}}{4-\left | x \right |}dx$$

REASON

$$\displaystyle \int_{-1}^{1}\left ( f\left ( x \right )+g\left ( x \right ) \right )dx=2\displaystyle \int_{0}^{1}f\left ( x \right )dx$$ if $$g(x)$$ is an odd function and $$f(x)$$ is an even function.

  • A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
  • B. Assertion is correct but Reason is incorrect
  • C. Both Assertion and Reason are incorrect
  • D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q4 Single Correct Hard
Evaluate the integral, $$\int _{ 0 }^{ 1 }{ \cos { \left( 2\cot ^{ -1 }{ \sqrt { \dfrac { 1-x }{ 1+x }  }  }  \right)  }  } dx=$$
  • A. $$1/2$$
  • B. $$0$$
  • C. $$1$$
  • D. $$-1/2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q5 Subjective Medium
Evaluate the following integral as limit of sum:
$$\displaystyle \int_{1}^{4}(x^2-x)dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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