What is the integral of e^{-2}sin(x/2) ?

The integral of e^{-2}sin(x/2) is -2/(e^2)cos(x/2)+C

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integral of e^{-2}sin(x/2)

\int e^{- 2} sin (\frac{x}{2}) d x

- \frac{2}{e ^{2}} cos (\frac{x}{2}) + C

Solution steps

\int e^{- 2} sin (\frac{x}{2}) d x

Take the constant out:

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

= e^{- 2} \cdot \int sin (\frac{x}{2}) d x

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{e ^{2}} \cdot \int sin (\frac{x}{2}) d x

Apply u-substitution

: \int sin (u) \cdot 2 d u

= \frac{1}{e ^{2}} \cdot \int sin (u) \cdot 2 d u

Take the constant out:

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

= \frac{1}{e ^{2}} \cdot 2 \cdot \int sin (u) d u

Use the common integral:

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

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

Substitute back

u = \frac{x}{2}

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

Simplify

\frac{1}{e ^{2}} \cdot 2 (- cos (\frac{x}{2})) : - \frac{2}{e ^{2}} cos (\frac{x}{2})

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

Add a constant to the solution

= - \frac{2}{e ^{2}} cos (\frac{x}{2}) + C

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What is the integral of e^{-2}sin(x/2) ?
The integral of e^{-2}sin(x/2) is -2/(e^2)cos(x/2)+C