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Question: How do you integrate \({e^{\sin x}}\cos xdx\) ?...

How do you integrate esin⁡xcos⁡xdx{e^{\sin x}}\cos xdx ?

Explanation

Solution

In this question, we have been asked to integrate the given expression. In order to solve this question, use a substitution method. After substituting, differentiate the equation and put it in the given expression. The, integrate the resultant expression and put the assumed value in the answer.

Complete step-by-step solution:
We are given an expression and we have been asked to integrate it.
⇒∫esin⁡xcos⁡xdx\Rightarrow \int {{e^{\sin x}}\cos xdx} …. (given)
We will solve this question using a substitution method as we can see that there is a trigonometric ratio in the power, whose differentiated value also exists in the given question itself. In such cases, we use substitution method
Let u=sin⁡xu = \sin x ……….…. (1)
Differentiating both the sides with respect to xx ,
dudx=cos⁡x\dfrac{{du}}{{dx}} = \cos x
Shifting the denominator to the other side,
du=cos⁡xdxdu = \cos xdx ………...…. (2)
Putting equation (1) and equation (2) in the given equation,
⇒∫eudu\Rightarrow \int {{e^u}du}
Now, we know that integration of ex{e^x} is ex{e^x} itself.
Using this,
⇒∫eudu=eu+C\Rightarrow \int {{e^u}du} = {e^u} + C
Now, we will substitute u=sin⁡xu = \sin x.
⇒esin⁡x+C\Rightarrow {e^{\sin x}} + C

Hence, ∫esin⁡xcos⁡xdx=esin⁡x+C\int {{e^{\sin x}}\cos xdx} = {e^{\sin x}} + C.

Note: In this question, we used the method of “Integration by Substitution”. This method is also known as the ‘u-Substitution’ or ‘The Reverse Chain Rule’ method. It is particularly used when the differentiation of the function, and the function – both are in the question together. It can be written as – ∫f(g(x))g′(x)dx\int {f\left( {g\left( x \right)} \right)g'\left( x \right)dx}
Here, we will substitute u=g(x)u = g\left( x \right) and then, we will differentiate it on both the sides with respect to x. We will get –
dudx=g′(x)⇒du=g′(x)dx\dfrac{{du}}{{dx}} = g'\left( x \right) \Rightarrow du = g'\left( x \right)dx
Hence, if we substitute the results in the equation, our question now becomes,
∫f(u)du\int {f\left( u \right)du} .
Thus, we have simplified our question. Make sure to find the answer in terms of x, and not in terms of u as u was not defined in the question and we generated it ourselves.