What is the integral of cos(x)e^{5+2sin(x)} ?

The integral of cos(x)e^{5+2sin(x)} is 1/2 e^{5+2sin(x)}+C

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integral of cos(x)e^{5+2sin(x)}

\int cos (x) e^{5 + 2 s i n (x)} d x

\frac{1}{2} e^{5 + 2 s i n (x)} + C

Solution steps

\int cos (x) e^{5 + 2 s i n (x)} d x

cos (x) e^{5 + 2 s i n (x)} = e^{5} cos (x) e^{2 s i n (x)}

= \int e^{5} cos (x) e^{2 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^{5} \cdot \int cos (x) e^{2 s i n (x)} d x

Apply u-substitution

= e^{5} \cdot \int \frac{e ^{u}}{2} d u

Take the constant out:

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

= e^{5} \frac{1}{2} \cdot \int e^{u} d u

Use the common integral:

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

= e^{5} \frac{1}{2} e^{u}

Substitute back

u = 2 sin (x)

= e^{5} \frac{1}{2} e^{2 s i n (x)}

Apply exponent rule:

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

e^{5} e^{2 s i n (x)} = e^{5 + 2 s i n (x)}

= \frac{1}{2} e^{5 + 2 s i n (x)}

Add a constant to the solution

= \frac{1}{2} e^{5 + 2 s i n (x)} + C

What is the integral of cos(x)e^{5+2sin(x)} ?
The integral of cos(x)e^{5+2sin(x)} is 1/2 e^{5+2sin(x)}+C