What is the integral of-4e^{5x} ?

The integral of-4e^{5x} is -4/5 e^{5x}+C

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integral of-4e^{5x}

\int - 4 e^{5 x} d x

- \frac{4}{5} e^{5 x} + C

Solution steps

\int - 4 e^{5 x} d x

Take the constant out:

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

= - 4 \cdot \int e^{5 x} d x

Apply u-substitution

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

= - 4 \cdot \int e^{u} \frac{1}{5} d u

Take the constant out:

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

= - 4 \cdot \frac{1}{5} \cdot \int e^{u} d u

Use the common integral:

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

= - 4 \cdot \frac{1}{5} e^{u}

Substitute back

u = 5 x

= - 4 \cdot \frac{1}{5} e^{5 x}

Simplify

- 4 \cdot \frac{1}{5} e^{5 x} : - \frac{4}{5} e^{5 x}

= - \frac{4}{5} e^{5 x}

Add a constant to the solution

= - \frac{4}{5} e^{5 x} + C

\int sin (2 x) sin (8 x) d x

y^{^{''}} + 5 y = 0, y^{^{'}} (1) = 3

x \to 0 + lim (tan (x))

\int \frac{1}{cos ( 2 y ) - cos ^{2} ( y )} d y

f (x) = (x - 2) (x^{2} + 4), (1, - 5)