Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

By seeing two functions multiplied, one of them being $e^x$, we are tempted to use Integration by parts method to solve it. Here also, our approach will be the same.
So, according to ILATE rule, our first function will be cos(x) , and second function will be $e^x$.
Let $I=\int e^{x}cos\left(x\right)dx$
$I=cos\left(x\right).e^x-\int e^x(-sin(x\left)\right)dx$
Or $I=cos\left(x\right).e^x+\int e^{x}sin\left(x\right)dx$
Let $I_1=\int e^{x}sin\left(x\right)dx$
$I=cos\left(x\right).e^x+I_1$……(1)
We can see that the integral is similar to the integral we are trying to solve. We’ll apply integration by parts methods here as well.
Here, sin(x) will be the first function here, and $e^x$ will be the second.
$I_1=sin\left(x\right).e^x-\int e^{x}cos\left(x\right)dx$
Let's substitute $I_1=sin\left(x\right).e^x-\int e^{x}cos\left(x\right)dx$ in the 1st equation.
So, $I=cos\left(x\right).e^x+sin\left(x\right).e^x-\int e^{x}cos\left(x\right)$
We can see that $\int e^{x}cos\left(x\right)$ in the above equation is nothing but I.
$I=cos\left(x\right).e^x+sin\left(x\right).e^x-I$
Or $2I=cos\left(x\right).e^x+sin\left(x\right).e^x$
$I=\frac{e^x\left(cos\right(x)+sin(x\left)\right)}{2}$