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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

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. $0$

1 Verified Answer | Published on 17th 09, 2020

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

1 Verified Answer | Published on 17th 09, 2020

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. $\displaystyle \frac {1}{2} \left[x^2F(x^2) - \int F(x^2) d(x^2) \right]$

1 Verified Answer | Published on 17th 09, 2020

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

1 Verified Answer | Published on 17th 09, 2020

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 $\sqrt{mx^2+nx+p}=(x-\alpha)u\:or\:(x-\beta)u$