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
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