# In the given equation as follows , use a table

In the given equation as follows , use a table of integrals with forms involving ln u to find the indefinite integral:-
$$\displaystyle\int{\left({\cos{{x}}}\right)}{e}^{{{\sin{{x}}}}}{\left.{d}{x}\right.}$$

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turtletalk75
Step 1
The given integral can be evaluated by the method of substitution.
We know that,
$$\displaystyle\int{e}^{{{x}}}{\left.{d}{x}\right.}={e}^{{{x}}}+{c}$$
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{x}}}={\cos{{x}}}$$
Step 2
The given integral is,
$$\displaystyle{I}=\int{\cos{{x}}}{e}^{{{\sin{{x}}}}}{\left.{d}{x}\right.}$$.
Put,
$$\displaystyle{\sin{{x}}}={u}$$
$$\displaystyle{d}{u}={\cos{{x}}}{\left.{d}{x}\right.}$$
Then,
$$\displaystyle{I}=\int{e}^{{{u}}}{d}{u}$$
$$\displaystyle={e}^{{{u}}}+{c}$$
$$\displaystyle={e}^{{{\sin{{x}}}}}+{c}$$
Hence, the required expression is, $$\displaystyle{e}^{{{\sin{{x}}}}}+{c}$$.
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Shawn Kim
Given:
$$\displaystyle\int{e}^{{{\sin{{x}}}}}{\cos{{\left({x}\right)}}}{\left.{d}{x}\right.}$$
Substitution $$\displaystyle{u}={\sin{{\left({x}\right)}}}\Rightarrow{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={\cos{{\left({x}\right)}}}$$
$$\displaystyle=\int{e}^{{{u}}}{d}{u}$$
$$\displaystyle\int{a}^{{{u}}}{d}{u}={\frac{{{a}^{{{u}}}}}{{{\ln{{\left({a}\right)}}}}}}$$ at a=e:
$$\displaystyle={e}^{{{u}}}$$
$$\displaystyle={e}^{{{\sin{{\left({x}\right)}}}}}$$
Result:
$$\displaystyle={e}^{{{\sin{{\left({x}\right)}}}}}+{C}$$
karton

$$\int(\cos(x))*e^{\sin(x)}dx$$
We put the expression $$\cos (x)$$ under the differential sign, i.e:
$$\cos(x)dx=d(\sin (x)), t=\sin(x)$$
Then the original integral can be written as follows:
$$\int e^{t}dt$$
This is a tabular integral:
$$\int e^{t}dt=e^{t}+C$$
To write the final The answer is, it remains to substitute $$\sin (x)$$ instead of t.
$$e^{\sin(x)}+C$$