Mathematics

# Integrate:$\displaystyle \int e^x\sin x\ dx$

##### SOLUTION

Consider the given integral.

$I=\int{{{e}^{x}}\sin xdx}$

We will use integration by parts method.

We know that,

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

Let,

$u={{e}^{x}}\Rightarrow du={{e}^{x}}dx$

$dv=\sin x\Rightarrow v=-\cos x$

Then,

$I={{e}^{x}}\left( -\cos x \right)-\int{{{e}^{x}}\left( -\cos x \right)dx}$

$I=-{{e}^{x}}\cos x+\int{{{e}^{x}}\cos xdx}$

Again, we will use integration by parts.

Let,

$p={{e}^{x}}\Rightarrow dp={{e}^{x}}dx$

$dq=\cos x\Rightarrow q=\sin x$

Then,

$I=-{{e}^{x}}\cos x+{{e}^{x}}\sin x-\int{{{e}^{x}}\sin xdx}+C$

$I=-{{e}^{x}}\cos x+{{e}^{x}}\sin x-I+C$

$2I={{e}^{x}}\left( \sin x-\cos x \right)+C$

$I=\dfrac{{{e}^{x}}}{2}\left( \sin x-\cos x \right)+C$

Hence, this is the required value of the integral.

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

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