Mathematics

# $\displaystyle \int \frac { e ^ { x } ( 1 + \sin x ) } { 1 + \cos x } d x =$

$e ^ { x } \tan \frac { x } { 2 } + c$

##### SOLUTION
Let $I=\displaystyle\int\dfrac{e^x(1+\sin x)}{1+\cos x}dx$

$=\displaystyle\int e^x\left(\dfrac{1+2\sin\dfrac{x}{2}\cos\dfrac{x}{2}}{2\cos^2\dfrac{x}{2}}\right)$

$=\displaystyle\int e^x\left(\dfrac{\sin^2\dfrac{x}{2}+\cos^2\dfrac{x}{2}+2\sin\dfrac{x}{2}.\cos\dfrac{x}{2}}{2\cos^2\dfrac{x}{2}}\right)$

$=\displaystyle\int e^x\left(\dfrac{\left(\sin\dfrac{x}{2}+\cos\dfrac{x}{2}\right)^2}{2\cos^2\dfrac{x}{2}}\right)$

$=\displaystyle\int \dfrac{e^x}{2}\left(\dfrac{\sin\dfrac{x}{2}+\cos\dfrac{x}{2}}{\cos\dfrac{x}{2}}\right)^2$

$=\displaystyle\int\dfrac{e^x}{2}\left(\dfrac{\sin\dfrac{x}{2}}{\cos\dfrac{x}{2}}+\dfrac{\cos\dfrac{x}{2}}{\cos\dfrac{x}{2}}\right)^2$

$=\displaystyle\int\dfrac{e^x}{2}\left(\tan\dfrac{x}{2}+1\right)^2$

$=\displaystyle\int\dfrac{e^x}{2}\left(\tan^2\dfrac{x}{2}+1+2\tan\dfrac{x}{2}\right)$

$=\displaystyle\int\dfrac{e^x}{2}\left(\sec^2\dfrac{x}{2}+2\tan\dfrac{x}{2}\right)$

$=\displaystyle\int e^x\left(\dfrac{1}{2}.\sec^2\dfrac{x}{2}+\dfrac{2}{2}\tan\dfrac{x}{2}\right)$

$=\displaystyle\int e^x\left(\tan\dfrac{x}{2}+\dfrac{1}{2}\sec^2\dfrac{x}{2}\right)$

Above is in the form of $e^x[f(x)+f'(x)]dx=e^xf(x)+c$
$=e^x\tan\dfrac{x}{2}+c$

$\therefore$  $\displaystyle\int\dfrac{e^x(1+\sin x)}{1+\cos x}dx$  $=e^x\tan\dfrac{x}{2}+c$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
If $\displaystyle \int_{0}^{1}xe^{x^{2}}dx= \lambda \int_{0}^{1}e^{x^{2}}dx$ then $\lambda$
• A. $\displaystyle \lambda = 0$
• B. $\displaystyle \lambda \epsilon = \left (-\infty, 0 \right )$
• C. $\displaystyle \lambda \epsilon = \left ( 1, 2 \right )$
• D. $\displaystyle \lambda \epsilon = \left ( 0,1 \right )$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Prove that:
$\displaystyle \int \dfrac {x^{2}dx}{(x\sin x+\cos x)^{2}}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\int_{}^{} {{{\tan }^3}2x\sec 2x{\rm{ dx}}}$ is equal to
• A. $\dfrac{1}{3}{\sec ^3}2x - \frac{1}{2}\sec 2x + c$
• B. $\dfrac{{ - 1}}{6}{\sec ^3}2x - \frac{1}{2}\sec 2x + c$
• C. $\dfrac{1}{3}{\sec ^3}2x + \frac{1}{2}\sec 2x + c$
• D. $\dfrac{1}{6}{\sec ^3}2x - \frac{1}{2}\sec 2x + c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle x^{x}(1+\log x)dx$ is equal to
• A. $x^{x}\log_{e}x+c$
• B. $e^{x^{x}}+c$
• C. $None\ of\ these$
• D. $x^{x}+c$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int_{0}^{1} \tan^{-1}\left( \dfrac { 2x-1 }{ 1+x-{ x }^{ 2 } } \right) dx=$
• A. $1$
• B. $\pi$
• C. $2\pi$
• D. $0$