Mathematics

# Write a value of $\int { { e }^{ x }\left( \sin { x } +\cos { x } \right) } dx$

##### SOLUTION
$I=\displaystyle\int{{e}^{x}\left(\sin{x}+\cos{x}\right)dx}$

It is of the form $\displaystyle\int{{e}^{x}\left(f\left(x\right)+{f}^{\prime}{\left(x\right)}\right)dx}={e}^{x}f\left(x\right)+c$

Put $f\left(x\right)=\sin{x}$ and ${f}^{\prime}{\left(x\right)}=\cos{x}$

$\therefore\,\displaystyle\int{{e}^{x}\left(\sin{x}+\cos{x}\right)dx}={e}^{x}\sin{x}+c$      ......where $c$ is the constant of integration.

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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
$\underset{-a}{\overset{a}{\int}} \sqrt{\dfrac{a - x}{a + x}} dx$ is equal to
• A. $\pi$
• B. a
• C. $\dfrac{a \pi}{2}$
• D. $a \pi$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Evaluate the following integrals:
$\int { \cfrac { { x }^{ 2 }({ x }^{ 4 }+4) }{ { x }^{ 2 }+4 } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
Integrate $\dfrac{\tan^4 \sqrt x+ \sec^2 \sqrt x}{\sqrt x}$
The solution is $\dfrac {2\tan ^3(\sqrt x)}{m}-2\tan \sqrt x+2\sqrt x+2\tan \sqrt x+C$.Find m

1 Verified Answer | Published on 17th 09, 2020

Q4 Multiple Correct Hard
If $\displaystyle I=\int e^{x\sin x+\cos x}\left ( \frac{x^{4}\cos^{3}x-x\sin x+\cos x}{x^{2}\cos ^{2}x} \right )dx$ then I equals

• A. $\displaystyle I= e^{x\sin x+\cos x}\left ( x\sin x-\frac{\cos x}{x} \right )$
• B. $\displaystyle I=e^{x\sin x+\cos x}\left ( \frac{x}{\tan x}-\frac{\cos x}{x} \right )+C$
• C. $\displaystyle I= e^{x\sin x+\cos x}\left ( x-\frac{\sec x}{x} \right )+C$
• D. $\displaystyle I=xe^{x\sin x+\cos x}-\int e^{x\sin x+\cos x}\left ( 1-\frac{\cos x-x\sin x}{x^{2}\cos^{2}x} \right )dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Evaluate:$\displaystyle \int \frac{dx}{\sqrt{(2-x)^{2}+1}}$
• A. $- \dfrac 12\log\left | (2-x)+\sqrt{x^{2}-4x+5} \right |+C$
• B. $-2\log\left | (2-x)+\sqrt{x^{2}-4x+5} \right |+C$
• C. none of these
• D. $-\log\left | (2-x)+\sqrt{x^{2}-4x+5} \right |+C$