What is the integral from-2 to 3 of e^xsin(-e^x) ?

The integral from-2 to 3 of e^xsin(-e^x) is cos(e^3)-cos(1/(e^2))

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integral from-2 to 3 of e^xsin(-e^x)

\int_{- 2}^{3} e^{x} sin (- e^{x}) d x

cos (e^{3}) - cos (\frac{1}{e ^{2}})

Decimal

- 0.66226 \dots

Solution steps

\int_{- 2}^{3} e^{x} sin (- e^{x}) d x

Simplify

e^{x} sin (- e^{x}) : - e^{x} sin (e^{x})

= \int_{- 2}^{3} - e^{x} sin (e^{x}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{- 2}^{3} e^{x} sin (e^{x}) d x

Apply u-substitution

= - \int_{\frac{1}{e ^{2}}}^{e^{3}} sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= - [- cos (u)]_{\frac{1}{e ^{2}}}^{e^{3}}

Compute the boundaries:

- cos (e^{3}) + cos (\frac{1}{e ^{2}})

= - (- cos (e^{3}) + cos (\frac{1}{e ^{2}}))

Simplify

= cos (e^{3}) - cos (\frac{1}{e ^{2}})

What is the integral from-2 to 3 of e^xsin(-e^x) ?
The integral from-2 to 3 of e^xsin(-e^x) is cos(e^3)-cos(1/(e^2))