What is the integral of ce^{cx} ?

The integral of ce^{cx} is e^{cx}+C

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integral of ce^{cx}

\int c e^{c x} d x

e^{c x} + C

Solution steps

\int c e^{c x} d x

Take the constant out:

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

= c \cdot \int e^{c x} d x

Apply u-substitution

: \int \frac{e ^{u}}{c} d u

= c \cdot \int \frac{e ^{u}}{c} d u

Take the constant out:

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

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

Use the common integral:

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

= c \frac{1}{c} e^{u}

Substitute back

u = c x

= c \frac{1}{c} e^{c x}

Simplify

c \frac{1}{c} e^{c x} : e^{c x}

= e^{c x}

Add a constant to the solution

= e^{c x} + C

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