What is the integral of 5e^{-x} ?

The integral of 5e^{-x} is -5e^{-x}+C

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integral of 5e^{-x}

\int 5 e^{- x} d x

- 5 e^{- x} + C

Solution steps

\int 5 e^{- x} d x

Take the constant out:

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

= 5 \cdot \int e^{- x} d x

Apply u-substitution

: \int - e^{u} d u

= 5 \cdot \int - e^{u} d u

Take the constant out:

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

= 5 (- \int e^{u} d u)

Use the common integral:

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

= 5 (- e^{u})

Substitute back

u = - x

= 5 (- e^{- x})

Simplify

= - 5 e^{- x}

Add a constant to the solution

= - 5 e^{- x} + C

y^{^{'}} - 4 x y = 0

y = 8 cos (x)

cos^{2} (x) sin (x) y^{^{'}} + (cos^{3} (x)) y = 1

(4 x^{2} + 5) (3 x - 2)

x \to 4 lim (\frac{a x ^{3}}{x - 4})