What is the integral of ae^{7x} ?

The integral of ae^{7x} is a 1/7 e^{7x}+C

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integral of ae^{7x}

\int a e^{7 x} d x

a \frac{1}{7} e^{7 x} + C

Solution steps

\int a e^{7 x} d x

Take the constant out:

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

= a \cdot \int e^{7 x} d x

Apply u-substitution

= a \cdot \int e^{u} \frac{1}{7} d u

Take the constant out:

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

= a \frac{1}{7} \cdot \int e^{u} d u

Use the common integral:

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

= a \frac{1}{7} e^{u}

Substitute back

u = 7 x

= a \frac{1}{7} e^{7 x}

Add a constant to the solution

= a \frac{1}{7} e^{7 x} + C