Lorraine Harvey

2022-01-05

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

turtletalk75

Step 1
The given integral can be evaluated by the method of substitution.
We know that,
$\int {e}^{x}dx={e}^{x}+c$
$\frac{d}{dx}\mathrm{sin}x=\mathrm{cos}x$
Step 2
The given integral is,
$I=\int \mathrm{cos}x{e}^{\mathrm{sin}x}dx$.
Put,
$\mathrm{sin}x=u$
$du=\mathrm{cos}xdx$
Then,
$I=\int {e}^{u}du$
$={e}^{u}+c$
$={e}^{\mathrm{sin}x}+c$
Hence, the required expression is, ${e}^{\mathrm{sin}x}+c$.

Shawn Kim

Given:
$\int {e}^{\mathrm{sin}x}\mathrm{cos}\left(x\right)dx$
Substitution $u=\mathrm{sin}\left(x\right)⇒\frac{du}{dx}=\mathrm{cos}\left(x\right)$
$=\int {e}^{u}du$
$\int {a}^{u}du=\frac{{a}^{u}}{\mathrm{ln}\left(a\right)}$ at a=e:
$={e}^{u}$
$={e}^{\mathrm{sin}\left(x\right)}$
Result:
$={e}^{\mathrm{sin}\left(x\right)}+C$

karton

$\int \left(\mathrm{cos}\left(x\right)\right)\ast {e}^{\mathrm{sin}\left(x\right)}dx$
We put the expression $\mathrm{cos}\left(x\right)$ under the differential sign, i.e:
$\mathrm{cos}\left(x\right)dx=d\left(\mathrm{sin}\left(x\right)\right),t=\mathrm{sin}\left(x\right)$
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 $\mathrm{sin}\left(x\right)$ instead of t.
${e}^{\mathrm{sin}\left(x\right)}+C$

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