Question

Choose the correct value of the given integral \int {{e^x}\left\\{ {\dfrac{{\sin x + \cos x}}{{1 - {{\sin }^2}x}}} \right\\}dx} from the below options.

1. ${e^x}\cos ecx + c$
2. ${e^x} + \cot x + c$
3. ${e^x}\sec x + c$
4. ${e^x}\tan x + c$

Answer 1

Hint : At first we need to simplify the given function to integrate. And then we try to integrate the given function. After simplifying the given expression we will end up with a small function in the form of $\sec x$. And then we are able to find a function whose differentiation is the given function.

Complete step-by-step answer :
Let us take,
f\left( x \right) = {e^x}\left\\{ {\dfrac{{\sin x + \cos x}}{{1 - {{\sin }^2}x}}} \right\\} ,
We know that ${\sin ^2}x + {\cos ^2}x = 1$ ,
By using the above identity we get,
\Rightarrow f\left( x \right) = {e^x}\left\\{ {\dfrac{{\sin x + \cos x}}{{{{\cos }^2}x}}} \right\\},
After simplification, we get
\Rightarrow f\left( x \right) = {e^x}\left\\{ {\dfrac{{\sin x}}{{{{\cos }^2}x}} + \sec x} \right\\},
Let us rename the trigonometric ratios,
\Rightarrow f\left( x \right) = {e^x}\left\\{ {\sec x.\tan x + \sec x} \right\\},
Take the ${e^x}$term inside the bracket
$\Rightarrow f\left( x \right) = {e^x}\sec x.\tan x + {e^x}\sec x$
Now take,
$g\left( x \right) = {e^x}\sec x$
Let us differentiate the above function.
${g^1}\left( x \right) = {e^x}\sec x + {e^x}\sec x\tan x$,
Observe that, $f\left( x \right) = {g^1}\left( x \right)$
$\Rightarrow \int {f\left( x \right)dx} = \int {{g^1}\left( x \right)dx}$,
After simplification, we get
$\Rightarrow \int {f\left( x \right)dx} = g\left( x \right) + c$
Now let us substitute the value of $g\left( x \right)$ , we get
$\Rightarrow \int {f\left( x \right)dx} = {e^x}\sec x + c$ .
Therefore the answer to the given integral is ${e^x}\sec x + c$
The correct option is 3.
So, the correct answer is “Option 3”.

Note : This is a very good problem. It is not easy to find the $g\left( x \right)$ in the first place but from practice, we can get the ideas quickly. After determining the value of $g\left( x \right)$ the rest of the problem is simple calculations. We can check the options to save time in the examination. And the simplification part is also slightly tricky, if we do the simplification part wrong there might be an error in the final result. If it is descriptive we may forget to add the integration constant. We have to keep an eye on the integration constant also. We used the basic trigonometric identities to simplify the given function.