What is the integral of 5e^{9x} ?

The integral of 5e^{9x} is 5/9 e^{9x}+C

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integral of 5e^{9x}

\int 5 e^{9 x} d x

\frac{5}{9} e^{9 x} + C

Solution steps

\int 5 e^{9 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^{9 x} d x

Apply u-substitution

: \int e^{u} \frac{1}{9} d u

= 5 \cdot \int e^{u} \frac{1}{9} d u

Take the constant out:

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

= 5 \cdot \frac{1}{9} \cdot \int e^{u} d u

Use the common integral:

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

= 5 \cdot \frac{1}{9} e^{u}

Substitute back

u = 9 x

= 5 \cdot \frac{1}{9} e^{9 x}

Simplify

5 \cdot \frac{1}{9} e^{9 x} : \frac{5}{9} e^{9 x}

= \frac{5}{9} e^{9 x}

Add a constant to the solution

= \frac{5}{9} e^{9 x} + C

f (x) = x^{4}, (2, 16)

x \to 0 lim (+ (x))

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

\frac{d}{d x} (\frac{e ^{2 x} - 1}{x})