What is the integral of e^{x/6+4} ?

The integral of e^{x/6+4} is 6e^{4+x/6}+C

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integral of e^{x/6+4}

\int e^{\frac{x}{6} + 4} d x

6 e^{4 + \frac{x}{6}} + C

Solution steps

\int e^{\frac{x}{6} + 4} d x

e^{\frac{x}{6} + 4} = e^{4} e^{\frac{x}{6}}

= \int e^{4} e^{\frac{x}{6}} d x

Take the constant out:

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

= e^{4} \cdot \int e^{\frac{x}{6}} d x

Apply u-substitution

: \int e^{u} \cdot 6 d u

= e^{4} \cdot \int e^{u} \cdot 6 d u

Take the constant out:

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

= e^{4} \cdot 6 \cdot \int e^{u} d u

Use the common integral:

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

= e^{4} \cdot 6 e^{u}

Substitute back

u = \frac{x}{6}

= e^{4} \cdot 6 e^{\frac{x}{6}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{4} e^{\frac{x}{6}} = e^{4 + \frac{x}{6}}

= 6 e^{4 + \frac{x}{6}}

Add a constant to the solution

= 6 e^{4 + \frac{x}{6}} + C

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