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

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

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integral of cos(x)e^{sin(x)+2}

\int cos (x) e^{s i n (x) + 2} d x

e^{2 + s i n (x)} + C

Solution steps

\int cos (x) e^{s i n (x) + 2} d x

cos (x) e^{s i n (x) + 2} = e^{2} cos (x) e^{s i n (x)}

= \int e^{2} cos (x) e^{s i n (x)} d x

Take the constant out:

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

= e^{2} \cdot \int cos (x) e^{s i n (x)} d x

Apply u-substitution

: \int e^{u} d u

= e^{2} \cdot \int e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= e^{2} e^{u}

Substitute back

u = sin (x)

= e^{2} e^{s i n (x)}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{2} e^{s i n (x)} = e^{2 + s i n (x)}

= e^{2 + s i n (x)}

Add a constant to the solution

= e^{2 + s i n (x)} + C

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