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

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

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integral of e^{cos(x)}(-sin(x))

\int e^{c o s (x)} (- sin (x)) d x

e^{c o s (x)} + C

Solution steps

\int e^{c o s (x)} (- sin (x)) d x

Simplify

= \int - e^{c o s (x)} sin (x) d x

Take the constant out:

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

= - \int e^{c o s (x)} sin (x) d x

Apply u-substitution

: \int - e^{u} d u

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

Take the constant out:

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

= - (- \int e^{u} d u)

Use the common integral:

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

= - (- e^{u})

Substitute back

u = cos (x)

= - (- e^{c o s (x)})

Simplify

= e^{c o s (x)}

Add a constant to the solution

= e^{c o s (x)} + C

\int (x^{2} - 3)^{5} x d x

\int - 4 - x^{2} d x

(x^{2} + 2 y^{2}) = x y y^{^{'}}

y^{^{'}} = \frac{y ^{3} - 3}{( 3 x y ^{2} )}

f (x) = \frac{5 x}{e ^{x}}

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