Mathematics

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


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

Hence, this is the answer.
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

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. $${1 \over 2}{a^2}{\cos ^{ - 1}}\left( {{{{x^2}} \over {{a^2}}}} \right) + {1 \over 2}\sqrt {{a^4} + {x^4} + c} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$2^{-n}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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

View Answer