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
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