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

Beginner2022-01-06Added 29 answers

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,${e}^{\mathrm{sin}x}+c$ .

The given integral can be evaluated by the method of substitution.

We know that,

Step 2

The given integral is,

Put,

Then,

Hence, the required expression is,

Shawn Kim

Beginner2022-01-07Added 25 answers

Given:

$\int {e}^{\mathrm{sin}x}\mathrm{cos}\left(x\right)dx$

Substitution$u=\mathrm{sin}\left(x\right)\Rightarrow \frac{du}{dx}=\mathrm{cos}\left(x\right)$

$=\int {e}^{u}du$

$\int {a}^{u}du=\frac{{a}^{u}}{\mathrm{ln}\left(a\right)}$ at a=e:

$={e}^{u}$

$={e}^{\mathrm{sin}\left(x\right)}$

Result:

$={e}^{\mathrm{sin}\left(x\right)}+C$

Substitution

Result:

karton

Expert2022-01-11Added 613 answers

We put the expression

Then the original integral can be written as follows:

This is a tabular integral:

To write the final The answer is, it remains to substitute

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