Question

integration of e^x cosnx dx

Answer 1

Answer :

We solve the integration problem using by parts method.

Now,

$I = \int {e}^{x} cosnx \: dx$

$= cosnx \int {e}^{x} dx - \int \{ \frac{d}{dx}(cosnx) \times \int {e}^{x} dx \} dx$

$= {e}^{x} cosnx + n \int {e}^{x} sinnx \: dx$

$= {e}^{x} cosnx + n \: sinnx \int {e}^{x} dx - n \int \{ \frac{d}{dx} (sinnx) \times \int {e}^{x} dx \} dx$

$= {e}^{x} cosnx + n \: {e}^{x} sinnx - {n}^{2} \int {e}^{x} cosnx \: dx + c$

where c is integral constant

$= {e}^{x} cosnx + n \: {e}^{x} sinnx - {n}^{2} I + c$

$\implies I + {n}^{2} I = {e}^{x} cosnx + n \: {e}^{x} sinnx + c$

$\implies ( {n}^{2} + 1 ) I = {e}^{x} cosnx + n \: {e}^{x} sinnx + c$

$\implies I = \frac{ {e}^{x} cosnx + n \: {e}^{x} sinnx + c}{ {n}^{2} + 1}$

$\implies \boxed{ \int {e}^{x} cosnx \: dx = \frac{ {e}^{x} cosnx + n \: {e}^{x} sinnx + c}{ {n}^{2} + 1}}$

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