In the given equation as follows , use a table

Lorraine Harvey 2022-01-05 Answered
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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Expert Answer

turtletalk75
Answered 2022-01-06 Author has 598 answers
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
Answered 2022-01-07 Author has 5181 answers
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:
\(\displaystyle={e}^{{{\sin{{\left({x}\right)}}}}}+{C}\)
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karton
Answered 2022-01-11 Author has 8659 answers

\(\int(\cos(x))*e^{\sin(x)}dx\)
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.
\(e^{\sin(x)}+C\)

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