Mathematics

Evaluate the following integrals
$$\int { \cfrac { 1 }{ \sqrt { 3 } \sin { x } +\cos { x }  }  } dx$$


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Subjective Medium Published on 17th 09, 2020
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Q1 Single Correct Medium
The value of the definite integral $$\int _{ 0 }^{ 1 }{ (e-1) } \sqrt { ln(1+(e-1)x) } +{ e }^{ { x }^{ 2 } })dx$$ is equal to 
  • A. $$1$$
  • B. $$e$$
  • C. $$e^2$$
  • D. $$0$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q2 Single Correct Medium
Integrate :  $$\displaystyle\int { \sqrt { 1+x\sqrt { 1+\left( 1+x \right) \sqrt { 1+\left( x+2 \right) \left( x+4 \right)  }  }  }  } dx$$
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  • B. $$\displaystyle\frac { { x }^{ 2 } }{ 2 } +C$$
  • C. $$x+C$$
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Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q3 Single Correct Hard
If $$\displaystyle \int { f(x) } dx=F(x)$$, then $$\displaystyle \int { { x }^{ 3 } } f\left( { x }^{ 2 } \right) dx$$ equals to
  • A. $$ \displaystyle \frac {1}{2} \left[x^2(F(x))^2 - \int (F(x))^2 dx \right]$$
  • B. $$ \displaystyle \frac {1}{2} \left[x^2F(x) - \frac{1}{2} \int (F(x))^2 dx \right]$$
  • C. $$ \displaystyle \frac {1}{2} \left[x^2F(x^2) + \int F(x^2) d(x^2) \right]$$
  • D. $$ \displaystyle \frac {1}{2} \left[x^2F(x^2) - \int F(x^2) d(x^2) \right]$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q4 Subjective Hard
Evaluate: $$\displaystyle \int \dfrac {1}{3 + 5\cos x}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q5 Passage Hard
Let us consider the integral of the following forms
$$f{(x_1,\sqrt{mx^2+nx+p})}^{\tfrac{1}{2}}$$
Case I If $$m>0$$, then put $$\sqrt{mx^2+nx+C}=u\pm x\sqrt{m}$$
Case II If $$p>0$$, then put $$\sqrt{mx^2+nx+C}=u\pm \sqrt{p}$$
Case III If quadratic equation $$mx^2+nx+p=0$$ has real roots $$\alpha$$ and $$\beta$$, then put $$\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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