Mathematics

# Integrate with respect to $x$.:$e^{x}\sin x$

##### SOLUTION
$\int e^x \sin x dx$

This can be solved by applying the concept of integration by parts

$\int{udv}=uv-\int{vdu}$

Take  $\sin x$ as $u$ and $e^x$ as $dv$, we get

$\Rightarrow \int{e^x \sin x}\space dx=I$     (let)  ....$(1)$

$\Rightarrow I=\sin x (e^x)-\int{\cos x \space e^x}dx$

Again apply integration by parts

$\Rightarrow I=\sin x \space e^x-(\cos x \space e^x-\int{(-\sin x)e^x}dx)$

$\Rightarrow I=\sin x \space e^x-(\cos x \space e^x+\int{(\sin x)e^x}dx)$

$\Rightarrow I=\sin x \space e^x-(\cos x \space e^x+I)$               (from $(1)$)

$\Rightarrow I=\sin x \space e^x-\cos x \space e^x-I$

$\Rightarrow 2I=\sin x \space e^x-\cos x \space e^x$

$\Rightarrow 2I=e^x(\sin x -\cos x)$

$\Rightarrow I=\dfrac{e^x(\sin x -\cos x)}{2}$

Therefore,  $\Rightarrow \int e^x \space \sin x\space dx=\dfrac{e^x(\sin x -\cos x)}{2}$

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

#### Realted Questions

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Find $\int\frac{1}{sin\, x \,cos^3x}dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int e^{sin^{-1}x}[1+\frac{x}{\sqrt{1-x^{2}}}]$ dx $=$
• A. $e^{sin^{-1}x}+c$
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• C. $x^{2}e^{sin^{1}X}+c$
• D. $xe^{sin^{-1}x}+c$

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Q3 Subjective Medium
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Q4 Single Correct Hard
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• C. $None\ of\ these$
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