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

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

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integral of 5e^{-5x}

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

- e^{- 5 x} + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{5} e^{u} d u

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

Take the constant out:

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

= 5 (- \frac{1}{5} \cdot \int e^{u} d u)

Use the common integral:

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

= 5 (- \frac{1}{5} e^{u})

Substitute back

u = - 5 x

= 5 (- \frac{1}{5} e^{- 5 x})

Simplify

5 (- \frac{1}{5} e^{- 5 x}) : - e^{- 5 x}

= - e^{- 5 x}

Add a constant to the solution

= - e^{- 5 x} + C

\frac{d}{d x} (arctan (\frac{1}{1 + x ^{2}}))

\int_{- 5}^{5} 4 (25 - x^{2}) d x

8 x - 5

\frac{d ^{4}}{d x ^{4}} (ln (1 + x))

y^{^{'}} + x = 0