What is the integral of ke^{3x} ?

The integral of ke^{3x} is k 1/3 e^{3x}+C

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integral of ke^{3x}

\int k e^{3 x} d x

k \frac{1}{3} e^{3 x} + C

Solution steps

\int k e^{3 x} d x

Take the constant out:

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

= k \cdot \int e^{3 x} d x

Apply u-substitution

= k \cdot \int e^{u} \frac{1}{3} d u

Take the constant out:

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

= k \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= k \frac{1}{3} e^{u}

Substitute back

u = 3 x

= k \frac{1}{3} e^{3 x}

Add a constant to the solution

= k \frac{1}{3} e^{3 x} + C

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