Mathematics

# Evaluate $\int{e^{x} \left(\dfrac{1+\sin x}{1+\cos x}\right)}$

##### SOLUTION
$\begin{array}{l} \int { { e^{ x } }\left( { \cfrac { { 1+\sin x } }{ { 1+\cos x } } } \right) dx } & \\ \Rightarrow \int { \left[ { \cfrac { 1 }{ { 1+\cos x } } +\cfrac { { \sin x } }{ { 1+\cos x } } } \right] { e^{ x } }dx } & \\ \Rightarrow \int { \left[ { \cfrac { 1 }{ { 2{ { \cos }^{ 2 } }\cfrac { x }{ 2 } } } +\cfrac { { 2\sin \cfrac { x }{ 2 } .\cos \cfrac { x }{ 2 } } }{ { 2{ { \cos }^{ 2 } }\cfrac { x }{ 2 } } } } \right] { e^{ x } }dx } & \\ \Rightarrow \int { \left[ { \cfrac { { { { \sec }^{ 2 } }\cfrac { x }{ 2 } } }{ 2 } +\tan \cfrac { x }{ 2 } } \right] { e^{ x } }dx } & \\ \Rightarrow \int { \left[ { f'\left( x \right) +f\left( x \right) } \right] { e^{ x } }dx } & \left[ { \int { { e^{ x } }\left[ { f\left( x \right) +f'\left( x \right) } \right] dx={ e^{ x } }f\left( x \right) +c } } \right] \\ \Rightarrow { e^{ x } }.f\left( x \right) +c & \\ { e^{ x } }.\tan \cfrac { x }{ 2 } +c. & \end{array}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
$\int {x\sqrt {{{{a^2} - {x^2}} \over {{a^2} + {x^2}}}dx = } }$
• A. ${1 \over 2}{\sin ^{ - 1}}\left( {{{{x^2}} \over {{a^2}}}} \right) + \sqrt {{a^4} + {x^4} + c}$
• B. ${1 \over 2}{a^2}{\sin ^{ - 1}}\left( {{{{x^2}} \over {{a^2}}}} \right) + {1 \over 2}\sqrt {{a^4} - {x^4} + c}$
• C. ${1 \over 2}{\cos ^{ - 1}}\left( {{{{x^2}} \over {{a^2}}}} \right) + {1 \over 2}\sqrt {{a^4} - {x^4} + c}$
• D. ${1 \over 2}{a^2}{\cos ^{ - 1}}\left( {{{{x^2}} \over {{a^2}}}} \right) + {1 \over 2}\sqrt {{a^4} + {x^4} + c}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve $\int { \left( x+2 \right) \left( x-1 \right) } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int { \sin ^{ 2 }{ \cfrac { x }{ 2 } } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle I=\int_{0}^{\pi/2} \cos^{n} x \sin^{n} x\:dx=\lambda \int_{0}^{\pi/2} \sin^{n} x\:dx$ then $\lambda$ equals
• A. $2^{-n+1}$
• B. $2^{-n-1}$
• C. $2^{-1}$
• D. $2^{-n}$

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