What is the integral of 5e^{3x+1} ?

The integral of 5e^{3x+1} is 5/3 e^{3x+1}+C

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integral of 5e^{3x+1}

\int 5 e^{3 x + 1} d x

\frac{5}{3} e^{3 x + 1} + C

Solution steps

\int 5 e^{3 x + 1} d x

Take the constant out:

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

= 5 \cdot \int e^{3 x + 1} d x

Apply u-substitution

= 5 \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

= 5 \cdot \frac{1}{3} \cdot \int e^{u} d u

Use the common integral:

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

= 5 \cdot \frac{1}{3} e^{u}

Substitute back

u = 3 x + 1

= 5 \cdot \frac{1}{3} e^{3 x + 1}

Simplify

5 \cdot \frac{1}{3} e^{3 x + 1} : \frac{5}{3} e^{3 x + 1}

= \frac{5}{3} e^{3 x + 1}

Add a constant to the solution

= \frac{5}{3} e^{3 x + 1} + C