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

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

integral of e^{5x}5

\int e^{5 x} 5 d x

e^{5 x} + C

Solution steps

\int e^{5 x} \cdot 5 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

= 5 \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

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

Use the common integral:

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

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

Substitute back

u = 5 x

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

Simplify

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

= e^{5 x}

Add a constant to the solution

= e^{5 x} + C

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