Mathematics

# Evaluate$\int { { e }^{ x }(\tan { x } +1)secx\quad dx}$

##### SOLUTION
$\begin{array}{c}I = \int {{e^x}\left( {\tan x + 1} \right)\sec xdx} \\ = \int {{e^x}\tan x\sec x} dx + \int {{e^x}\sec x} dx\end{array}$

According to the integration by parts,
$\begin{array}{l}\int {uvdx} = u\int {vdx} - \int {\left( {\frac{{du}}{{dx}} \times \int {vdx} } \right)} dx\\Here,\\u = \sec x\\v = {e^x}\\So,\\\int {\sec x{e^x}dx} = \sec x\int {{e^x}dx} - \int {\left( {\frac{{d\sec x}}{{dx}} \times \int {{e^x}dx} } \right)} dx\\ = {e^x}\sec x - \int {\left( {\sec x\tan x{e^x}} \right)} dx\\So,\\I = \int {\left( {\sec x\tan x{e^x}} \right)} dx + {e^x}\sec x - \int {\left( {\sec x\tan x{e^x}} \right)} dx\\ = {e^x}\sec x\end{array}$

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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 Medium
Evaluate $\displaystyle\int^{\pi/2}_0\sin^2xdx$
• A. $\dfrac{\pi}{3}$
• B. $\dfrac{\pi}{2}$
• C. $\dfrac{2\pi}{3}$
• D. $\dfrac{\pi}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
If $f(x)=\dfrac{4}{\pi}\sin\left(\dfrac{\pi}{2}x\right)+B$ and $\int^0_1f(x)dx=\cfrac{4}{\pi}\int\sin(\cfrac{\pi}{2}x)+B dx$, Find $B$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
If $\int \dfrac {1}{1 + \sin x}dx = \tan \left (\dfrac {x}{2} + a\right ) + b$, then
• A. $a = -\dfrac {\pi}{4}, b\in R$
• B. $a = \dfrac {5\pi}{4}, b\in R$
• C. None of these
• D. $a = \dfrac {\pi}{4}, b\in R$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Integrate:
$\int _{ -\pi }^{ \pi }{ \dfrac { 2x(1+\sin { x } ) }{ 1+{ cos }^{ 2 }x } } { dx }\\$ is?
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Evaluate $\int \dfrac{e^x-e^{-x}}{e^x+e^{-x}}dx$