Question

How do you integrate ${e^{\sin x}}\cos xdx$?

Answer 1

In solving the given question, we have to integrate the given function, we first consider one part of the question as a variable, here it will be $u = {e^{\sin x}}$, now derive the function using the identity, $\dfrac{d}{{dx}}{e^x} = {e^x}$ and we will derive the function $\sin x$ again then we will get the second part of the function. Now we will get a function which will be easily integrated and that will be our required result.

Complete step-by-step answer:
Given function is ${e^{\sin x}}\cos xdx$,
We first consider $u = {e^{\sin x}}$,
Now differentiating on both sides we get,
\Rightarrow $$$$\dfrac{d}{{dx}}u = \dfrac{d}{{dx}}{e^{\sin x}},
Now using the identity $\dfrac{d}{{dx}}{e^x} = {e^x}$ and if the expression has two functions then we have to again derive the second function also, we get,
\Rightarrow $$$$\dfrac{{du}}{{dx}} = {e^{\sin x}}\dfrac{d}{{dx}}\sin x,
Now using the identity $\dfrac{d}{{dx}}\sin x = \cos x$, we get,
\Rightarrow $$$$\dfrac{{du}}{{dx}} = {e^{\sin x}}\cos x,
Now taking$dx$ to the right hand side of the expression we get,
\Rightarrow $$$$du = {e^{\sin x}}\cos xdx,
Now applying integration on both sides we get,
$\Rightarrow \int {{e^{\sin x}}\cos xdx} = \int {du}$,
Now using the integral identity $\int {dx = x + c}$, we get,
$\Rightarrow \int {{e^{\sin x}}\cos xdx} = u + c$,
Now substituting u value i.e.,$u = {e^{\sin x}}$ as we first considered, in the above expression we get,
$\Rightarrow \int {{e^{\sin x}}\cos xdx} = {e^{\sin x}} + c$,
So, the integral of ${e^{\sin x}}\cos xdx$is equal to${e^{\sin x}} + c$.

$\therefore$The integral value of ${e^{\sin x}}\cos xdx$ will be equal to ${e^{\sin x}} + c$.

Note:
In this type of question we use both derivation and integration formulas as both are related to each other so students should not get confused where to use the formulas and which one to use as there are many formulas in both derivation and integration. Some of the important formulas that are used are given below:
$\dfrac{d}{{dx}}x = 1$,
$\dfrac{d}{{dx}}{x^n} = n{x^{n - 1}}$,
$\dfrac{d}{{dx}}uv = u\dfrac{{dv}}{{dx}} + v\dfrac{{du}}{{dx}}$,
$\dfrac{d}{{dx}}{e^x} = {e^x}$,
$\dfrac{d}{{dx}}\dfrac{u}{v} = \dfrac{{v\dfrac{{du}}{{dx}} - u\dfrac{{dv}}{{dx}}}}{{{v^2}}}$,
$\dfrac{d}{{dx}}\ln x = \dfrac{1}{x}$,
$\dfrac{d}{{dx}}\sin x = \cos x$,
$\dfrac{d}{{dx}}\cos x = - \sin x$,
$\int {dx = x + c}$,
$\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + c$,
$\int {\dfrac{1}{x}dx} = \ln x + c$.