What is the integral of 1/4 e^{x/3} ?

The integral of 1/4 e^{x/3} is 3/4 e^{x/3}+C

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integral of 1/4 e^{x/3}

\int \frac{1}{4} e^{\frac{x}{3}} d x

\frac{3}{4} e^{\frac{x}{3}} + C

Solution steps

\int \frac{1}{4} e^{\frac{x}{3}} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int e^{\frac{x}{3}} d x

Apply u-substitution

= \frac{1}{4} \cdot \int e^{u} \cdot 3 d u

Take the constant out:

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

= \frac{1}{4} \cdot 3 \cdot \int e^{u} d u

Use the common integral:

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

= \frac{1}{4} \cdot 3 e^{u}

Substitute back

u = \frac{x}{3}

= \frac{1}{4} \cdot 3 e^{\frac{x}{3}}

Simplify

\frac{1}{4} \cdot 3 e^{\frac{x}{3}} : \frac{3}{4} e^{\frac{x}{3}}

= \frac{3}{4} e^{\frac{x}{3}}

Add a constant to the solution

= \frac{3}{4} e^{\frac{x}{3}} + C

What is the integral of 1/4 e^{x/3} ?
The integral of 1/4 e^{x/3} is 3/4 e^{x/3}+C